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WORKS OF 
PROFESSOR CECIL H. PEABODY 

PUBLISHED BY 

JOHN WILEY & SONS. 



Thermodynamics of the Steam=engine and other 
Heat-engines. 

This work is intended for the use of students in 
technical schools, and gives the theoretical training 
required by engineers. 8vo, cloth, $5.00. 

Tables of the Properties of Saturated Steam and 
other Vapors. 

These tables were prepared for the use of students 
in technical schools and colleges, and of engineers in 
general. 8vo, cloth, $1.00. 



Valve-gears for Steam-engines. 

This book is intended to give engineering students 
instruction in the theory and practice of designing 
valve-gears for steam-engines. 8vo, cloth, $2.50. 



Steam-boilers. 

By Prof. Cecil H. Peabody and 



F. Miller. 
cloth, S4.00. 



Prof. Edward 
Nearly 400 pages; 142 illustrations. 8vo, 



THERMODYNAMICS 



OF THE 



STEAM-ENGINE 



AND OTHER HEAT-ENGINES. 



BY 



CECIL H. PEABODY, 

Professor of Marine Engineering and Naval Architecture, 
Massachusetts Institute of Technology. 



FOURTH EDITION, REWRITTEN AND RESET. 
FIRST THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

London : CHAPMAN & HALL, Limited. 

18980 






14026 



Copyright, 1898, 

BY 

CECIL K. PEABODY. 





TWOCOprtTRtCtTvED. 

ndCOFY, 
1898. 



ROBERT DRUMMOND, PRINTER. NEW YORK. 




- Si > 



PREFACE. 



This work is designed to give instruction to students in 
technical schools in the methods and results of the application 
of thermodynamics to engineering. While it has been con- 
sidered desirable to follow commonly accepted methods, some 
parts differ from other text-books, either in substance or in 
manner of presentation, and may require a few words of 
explanation. 

The general theory or formal presentation of thermody- 
namics is that employed by the majority of writers, and was 
prepared with the view of presenting clearly the difficulties 
inherent in the subject, and of giving familiarity with the 
processes employed. 

In the discussion of the properties of gases and vapors the 
original experimental data on which the working equations, 
whether logical or empirical, must be based are given quite 
fully, to afford an idea of the degree of accuracy attainable in 
calculations made with their aid. Rowland's determination 
of the mechanical equivalent of heat has been adopted, and 
with it his determination of the specific heat of water at low 
temperatures. The author's " Tables of the Properties of 
Saturated Steam and Other Vapors' were calculated to 
accompany this work, and may be considered to be an 
integral part of it. 

The chapters on the flow of gases and vapors and on the 
injector are believed to present some novel features, espe- 
cially in the comparisons with experiments. 

The feature in which this book differs most from similar 
works is in the treatment of the steam-engine. It has been 
deemed advisable to avoid all approximate theories based on 

iii 



IV PREFA CE. 

the assumption of adiabatic changes of steam in an engine 
cylinder, and instead to make a systematic study of steam- 
engine tests, with the view of finding what is actually known 
on the subject, and how future investigations and improve- 
ments may be made. For this purpose a large number of 
tests have been collected, arranged, and compared. Special 
attention is given to the investigations of the action of steam 
in the cylinder of an engine, considerable space being given 
to Hirn's researches and to experiments that provide the 
basis for them. Directions are given for testing engines, and 
for designing simple and compound engines. 

Chapters have been added on compressed-air and refriger- 
ating machines, to provide for the study of these important 
subjects in connection with the theory of thermodynamics. 

Wherever direct quotations have been made, references 
have been given in foot-notes, to aid in more extended in- 
vestigations. It does not appear necessary to add other 
acknowledgment of assistance from well-known authors, 
further than to say that their writings have been diligently 
searched in the preparation of this book, since any text-book 
must be largely an adaptation of their work to the needs of 
instruction. 

C. H. P. 

Massachusetts Institute of Technology, 
May, 1889. 



PREFACE TO FOURTH EDITION. 



A THOROUGH revision of this work has been made to 
bring it into accord with more recent practice and to include 
later experimental work. Advantage is taken of this oppor- 
tunity to make changes in matter or in arrangement which it 
is believed will make it more useful as a text-book. 

C. H. P. 
Massachusetts Institute of Technology. 
July, 1898. 



TABLE OF CONTENTS. 



CHAPTER PAGE 

I. Thermal Capacities r 

II. First Law of Thermodynamics 15 

III. Second Law of Thermodynamics 25 

IV. Fundamental Equations 44 

V. Perfect Gases 54 

VI. Saturated Vapors 80 

VII. Superheated Vapors 123 

VIII. Flow of Fluids 149 

IX. Injectors 163 

X. Hot-air Engines and Gas-engines 194 

XI. The Steam-engine 229 

XII. Compound-engines 255 

XIII. Testing Steam-engines 280 

XIV. Influence of the Cylinder- walls 301 

XV. Economy of Steam-engines 353 

XVI. Friction of Engines 429 

XVII. Compressed Air 442 

XVIII. Refrigerating Machines , 479 

v 



THERMODYNAMICS OF THE STEAM-ENGINE. 



CHAPTER I. 
THERMAL CAPACITIES. 

THE object of thermodynamics, or the mechanical theory 
of heat, is the solution of problems involving the action of 
heat, and, for the engineer, more especially those problems 
presented by the steam-engine and other thermal motors. In 
this work the discussion of the actual nature of heat and the 
rationale of its various actions will be purposely avoided, and 
attention will be given rather to the calculation of the results 
of such actions. 

Effects of Heat. — In general the action of heat on a 
given substance changes all the characteristics of that sub- 
stance, such as density, temperature, elasticity, conductivity, 
etc. A comprehensive theory of thermodynamics should 
make it possible to calculate such changes in any of the char- 
acteristics of any substance. In fact, the number of substances 
for which we have adequate theoretical and experimental 
knowledge is limited, and only a few characteristics of such 
substances are commonly included in our discussions. 

The substances in which the engineer has the most interest 
are gases and vapors, more especially air and steam. Fortu- 
nately an adequate treatment can be given of these substances 
for engineering purposes. 



2 THERMODYNAMICS OF THE STEAM-ENGINE. 

First General Principle. — In the development of the 
theory of thermodynamics it is assumed that if any two 
characteristics or properties of a substance are known these 
two, treated as independent variables, will enable us to calcu- 
late any third property. 

As an example we have from the combination of the laws 
of Boyle and Gay-Lussac the general equation for gases. 

pv = RT, (i) 

in which / is the pressure, v is the volume, T is the absolute 
temperature by the air-thermometer, and R is a constant 
which for air has the value 53- 22 when English units are 
used. It is probable that this equation led to the general 
assumption just quoted. That assumption is purely arbitrary, 
and is to be justified by its results. It may properly be con- 
sidered to be the first general principle of the theory of ther- 
modynamics ; the other two general principles are the so- 
called first and second laws of thermodynamics, which will be 
stated and discussed later. 

Characteristic Equation. — An equation which gives the 
relations of the properties of any substance is called the 
characteristic equation for that substance. The properties 
appearing in a characteristic equation are commonly pressure, 
volume, and temperature, but other properties may be used 
if convenient. The form of the equation must be determined 
from experiments, either directly or indirectly. 

The characteristic equation for a gas is, as already quoted, 

pv == RT, 
which may be written also 

RT RT j.,,— 

p = — , v — — , pv — RT = o. 
r v P 

A similar treatment may be applied to any characteristic 



THERMAL CAPACITIES. 3 

equation in terms of pressure, volume, and temperature, so 
that we may have 

T = F(p, v), p = F X (T, v), v = FIT, p), 
or in general 

/(/, V, T) = (2) 

If x, y, and z represent any properties or characteristics of 
a substance, then the first general principle may be expressed 
algebraically by 

f(x, y, z) = o (3) 

Specific Pressure. — The pressure is assumed to be a 
hydrostatic pressure, such as a fluid exerts on the sides of the 
containing vessel or on an immersed body. The pressure is 
consequently the pressure exerted by the substance under con- 
sideration rather than the pressure on that substance. For 
example, in the cylinder of a steam-engine the pressure of the 
steam is exerted on the piston during the forward stroke and 
does work on the piston ; during the return stroke, when the 
steam is expelled from the cylinder, it still exerts pressure on 
the piston and abstracts work from it. 

For the purposes of the general theory pressures are 
expressed in terms of pounds on the square foot for the 
English system of units. In the metric system the pressure is 
expressed in terms of kilograms on the square metre. A 
pressure thus expressed is called the specific pressure. In 
engineering practice other terms are used, such as pounds on 
the square inch, inches of mercury, millimetres of mercury, 
atmospheres, or kilograms on the square centimetre. 

Specific Volume. — It is convenient to deal with one unit 
of weight of the substance under discussion, and to consider 
the volume occupied by one pound or one kilogram of the 
substance ; this is called the specific volume, and is expressed 
in cubic feet or in cubic metres. The specific volume of air 



4 THERMODYNAMICS OF THE STEAM-ENGINE. 

at freezing-point and under the normal atmospheric pressure 
is 12.39 cubic feet; the specific volume of saturated steam at 
212° F. is 26.6 cubic feet; and the specific volume of water 

is about , or nearly 0.O16 of a cubic foot. 

62.4' y 

Temperature is commonly measured by aid of a mercurial 
thermometer which has for its reference-points the freezing- 
point and boiling-point of water. A centigrade thermometer 
has the volume of the stem between the reference-points 
divided into one hundred equal parts called degrees. The 
Fahrenheit thermometer differs from the centigrade in having 
one hundred and eighty degrees between the freezing-point 
and the boiling-point, and in having its zero thirty-two degrees 
below freezing. 

The scale of a mercurial thermometer is entirely arbitrary, 
and its indications depend on the relative expansion of glass 
and mercury. Indications of such thermometers, however 
carefully made, differ appreciably, mainly on account of the 
varying nature of the glass. For refined investigations ther- 
mometry readings are reduced to the air-thermometer, which 
has the advantage that the expansion of air is so large com- 
pared with the expansion of glass that the latter has little or 
no effect. 

It is convenient in making calculations of the properties of 
air to refer temperatures to the absolute zero of the scale 
of the air-thermometer. To get a conception of what is 
meant by this expression we may imagine the air-thermom- 
eter to be made of a uniform glass tube with a proper 
index to show the volume of the air. The position of 
the index may be marked at boiling-point and at freez- 
ing-point as on the mercurial thermometer, and the space 
between may be divided into one hundred parts or degrees. 
If the graduations are continued to the closed end of the 
tube there will be found to be between 273 and 274 of 
them. It will be shown later that there is reason to suppose 
that the absolute zero of temperature is 273.7 degrees centi- 



THERMAL CAPACITIES. 5 

grade below the freezing-point of water. Speculations as to 
the meaning of absolute zero and discussions concerning the 
nature of substances at that temperature are not now profit- 
able. It is sufficient to know that equations are simplified 
and calculations are facilitated by this device. For example, 
if temperature is reckoned from the arbitrary zero of the 
centigrade thermometer, then the characteristic equation for a 
perfect gas becomes 

* = £ + *)* 

in which a is the coefficient of dilatation and - = 273.7 

nearly. 

In order to distinguish the absolute temperature from the 
temperature by the thermometer we shall designate the 
former by T and the latter by t, bearing in mind that 

T = t -f- 273. 7 centigrade, 
T = t + 460. 7 Fahrenheit. 

It will appear in the course of the development of the 
theory of thermodynamics that a scale of temperature can be 
constructed depending on the fundamental units of length and 
weight, such as the foot and the pound. Such a scale is 
properly called the absolute scale of temperature, because it 
does not depend on the properties of any substance (glass, 
mercury, or air), and because degrees may be given the same 
value or significance in all parts of the scale. That a degree 
on the air-thermometer has not the same value in all parts of 
the scale is shown by the fact that the scale of the air-ther- 
mometer differs slightly from that of the absolute thermom- 
eter, as will be seen from the table on page 73. The irregu- 
larities of the scale of a mercurial thermometer are much 
greater, so that physical observations are reduced to the scale 
of the air-thermometer; in engineering tests it is usually suffi- 



6 



THERMODYNAMICS OF THE STEAM-ENGINE. 



cient to take the readings of the mercurial thermometer with- 
out such a reduction. 

In the development of the theory of thermodynamics it will 
be assumed that temperatures are referred to the absolute scale, 
though as yet we do not know the nature of that scale or how 
it is constructed. The apparent indefiniteness accompanying 
this suspension of judgment is much more than compensated 
for by the ease with which the absolute scale can be defined 
when we arrive at the proper place for doing so. 

Graphical Representation of the Characteristic Equa- 
tion. — Any equation with three variables may be represented 
by a geometrical surface referred to coordinate axes, of which 
surface the variables are the coordinates. In the case of 
a perfect gas which conforms to the equation 

pv — RT 



the surface is such that each section perpendicular to the axis 

of T is a rectangular hyperbola (Fig. i). 
Returning now to the general case, 
it is apparent that the characteristic 
equation of any substance may be 
represented by a geometrical surface 
referred to coordinate axes, since the 
equation is assumed to contain only 
three variables; but the surface will in 
general be less simple in form than that 
representing the combined laws of Boyle and Gay-Lussac. 

If one of the variables, as T, is given a special constant 
value it is equivalent to taking a section perpendicular to the 
axis of T) and a plane curve will be cut from the surface, 
which may be conveniently projected on the (p, v) plane. 
The reason for choosing the (/, v) plane is that the curves 
correspond with those drawn by the steam-engine indicator. 

Considerable use is made of such thermal curves in explain- 
ing thermodynamic conceptions. As a rule, a graphical proc- 




THERMAL CAPACITIES. 7 

ess or representation is merely another way of presenting an 
idea that has been, or may be, presented analytically; there 
is, however, an advantage in representing a condition or 
a change to the eye by a diagram,, especially in a discussion 
which appears to be abstract. A number of thermal curves 
are explained on page 18. 

Standard Temperature. — For many purposes it is con- 
venient to take the freezing-point of water for the standard 
temperature, since it is one of the reference-points on the 
thermometric scale ; this is especially true for air. But the 
properties of water change rapidly at and near freezing-point 
and are very imperfectly known. It has consequently become 
customary to take 62 F. for the standard temperature for the 
English system of units ; there is a convenience in this, 
inasmuch as the pound and yard are standards at that tempera- 
ture. For the metric system 15 C. is used, though the kilo- 
gram and metre are standards at freezing-point. 

Thermal Unit. — Heat is measured in calories or in British 
thermal units (b. T. u.). A British thermal unit is the heat 
required to raise one pound of water from 62 F. to 63 F. ; 
in like manner a calorie is the heat required to raise one kilo- 
gram of water from 15 C. to 16 C. 

The calorie is often defined as the heat required to raise a 
kilogram of water from freezing-point to one degree centigrade ; 
and a B. T. u. is correspondingly defined as the heat required 
to raise one pound of water from freezing-point to 33 F. The 
objection to the use of the freezing-point as the standard tem- 
perature — namely, the uncertainty as to the properties of water 
at that temperature — applies even more forcibly here. The 
whole subject will be discussed later in connection with Row- 
land's determination of the mechanical equivalent of heat. 

The thermal unit, or the calorie, should depend on the 
absolute scale of temperature, but for practical purposes the 
scale of the air-thermometer is sufficient. 

Thermal Capacities. — The amount of heat required to 
change by unity any quality of a unit of weight of a substance 



8 THERMODYNAMICS OF THE STEAM-ENGINE. 

is called the thermal capacity corresponding to the given 
change. 

Three thermal capacities have received names, i. e., specific 
heat at constant volume, specific heat at constant pressure, 
and latent heat of expansion. 

Specific Heat is the number of thermal units required to 
raise a unit of weight of a given substance one degree of 
temperature. The specific heat of water at the standard tem- 
perature is, of course, unity. 

If the specific heat of a given substance is constant, then 
the heat required to raise one pound through a given range of 
temperature is the product of the specific heat by the increase 
of temperature. Thus if c is the specific heat and t — t x is the 
range of temperature the heat required is 

Q=c(t- /,), and c = ^ • 

If the specific heat varies the amount of heat must be 
obtained by integration — that is, 

Q =fcdt, 

and conversely 

c _dQ 
C ~~ dt 

It is customary to distinguish two specific heats for perfect 
gases; specific heat at constant pressure and specific heat at 
constant volume, which may be represented by 

fdQ\ i (dQ\ 

c*= -77 I , and c = — 1 . 



'* 



\dt); v ~~\dt): 



The subscript attached to the parenthesis indicates the 
property which is constant during the change. 

It is evident that the specific heats just expressed are 
partial differential coefficients. Partial differentials may often 



THERMAL CAPACITIES. Q 

be recognized by their positions in equations or by the con- 
text. Sometimes they are indicated by parentheses, as 
above, but without the subscripts ; or they may be indicated 
by a special type, as 

dQ 

dt' 

In thermodynamics several variables are often intro- 
duced, each of which may be a function of two independent 
variables; it is consequently convenient, if not essential, to 
indicate a partial differential by a parenthesis and a subscript, 
as above. 

Latent Heat of Expansion is the amount of heat 
required to increase the volume of a unit of weight of the 
substance by one cubic foot, or one cubic metre, at constant 
temperature. It may be represented by 

dQ\. 

dvjt " 

General Equations of the Effects Produced by Heat — 

In conformity with the first general principle the heat required 
to produce a change in a unit of weight of a given substance 
may be expressed as a function of any two properties of the 
substance ; thus we may have 

Q = F& v), Q = F a (t,p), or Q = F 3 (/>, v), . (4) 

Differentiating the several equations (4), we have 

iM^).£+(Sf)*» * ' ' * (5 " } 

,d0\ 4Q\ 



IO THERMODYNAMICS OF THE STEAM-ENGINE. 

l d Q\ 

In equation (^d) the partial differential coefficients '~J7j 

an d / _~\ are the specific heat at constant volume and the 

\dv) t 

latent heat of expansion ; they may be replaced by c P and /, 
giving 

dQ = c v dt + Idv (5) 

( d Q\ 
In like manner the differential coefficient f — jr ) in equa- 
tion (6a) may be replaced by c p . The differential coefficient 
(-^-) is also a thermal capacity, and represents the amount 

of heat that must be added to increase the pressure to the 
extent of one pound per square foot (or one kilogram per 
square metre). No name has been given to this thermal 
capacity ; it may be represented by the letter m, and equation 
(6a) becomes 

dQ = c p dt + mdp (6) 

Finally, l- r -\ may be represented by n, and f—\ by o y 
\dpj v \dvjp 

both being thermal capacities without names, and equation 

(yd) becomes 

dQ =z ndp -\- odv (7) 

Relations of the Thermal Capacities. — The three equa- 
tions (5), (6), and (7) show the changes produced by the 
addition of an amount of heat dQ to a unit of weight of a 
substance, the difference coming from the methods of analyz- 
ing the changes. We may conveniently find the relations of 
the several thermal capacities by the method of undetermined 
coefficients. Thus equating the right-hand members of equa- 
tions (5) and (6), 

c v dt -\- Idv = c p dt -\- mdp. . . . . (8) 



THERMAL CAPACITIES. II 

From the first general principle we have 

v = F(pT), 
from which, by differentiating we have 

which substituted in (8) gives 

c p dt + mdp = c v dt + /[( *\* + (g) #]. 

. •. c p dt + *k^ = p w + l(j\ dt + ^W- • • (9) 

It will be noted that, as T differs from t only by the addi- 
tion of a constant, the differential dt may be used in all cases, 
whether we are dealing with absolute temperatures or temper- 
atures on the ordinary thermometer. 

In equation (9) p and T are independent variables, and 
each may have all possible values; consequently we may 
equate like coefficients. 

■'■ c * = c ' +l $) f (I0) 

$),=*"* (II) 

Again, equating the remaining coefficients, 

l (i)r m (I2) 

Again, we have 

p = FIT, v), 
from which 

d p = (4?\ dt + {-P\ dv > • - ' ( J 3) 

\dt L \dv h 



12 THERMODYNAMICS OF THE STEAM-ENGINE. 

which substituted in (8) gives 

c p dt + ™[&) dt + $\ dv\ = c v dt + Idv, 

Equating like coefficients, 

c ' +m $)r c " (I4) 

or 

fdp 



(4\= c *-c v (15) 



©.. 



From equations (6) and (7) 

c p dt -f- mdp = ndp -\- odv (16) 

And from the equation 

T=F^p) 



we have 



— (!) *+(*).* 



which substituted in equation (16) gives 



Equating coefficients of dv, 

Finally, from equations (5) and (7) we have 
c v dt -\- Idv = ndp -\- odv. 



° = c >(jX (l7 > 



THERMAL CAPACITIES. 1 3 

Substituting for the value of dt, as above, 
(dt\ T . /dt 

V 

Equating coefficients of dp, 






fdt\ 
n = c. 



•(I) <■*> 



For convenience the several relations of the thermal ca- 
pacities may now be assembled as follows : 



*$i t =e '~' n 


- m (t)r^~ c " 


n=c -0: 


idt\ 


m 


= i(~\ . 



\dph 

They are the necessary algebraic relations of the literal 
functions growing out of the first general principle, and are 
independent of the scale of temperature, or of any other theo- 
retical or experimental principle of thermodynamics other 
than the one already stated — namely, that any two properties 
of a given substance, treated as independent variables, are 
sufficient to allow us to calculate any third property. 

Of the six thermal capacities the specific heat at constant 
pressure is the only one that is commonly known by direct 
experiment. For perfect gases this thermal capacity is a con- 
stant, and, further, the ratio of the specific heats 



is a constant, so that <r v is readily calculated. The relations of 
the thermal capacities allow us to calculate values for the 
other thermal capacities, /, m, n, and o, provided that we can 



14 THERMODYNAMICS OF THE STEAM-ENGINE. 

first determine the several partial differential coefficients which 
appear in the proper equations. But for a perfect gas the 
characteristic equation is 



from which we have 



consequently 



pv = RT, ...... (19) 



(dv\ R 



/=|ta-o> ...... (21) 



'so that / may be readily calculated for any pressure. Other 
partial differential coefficients can be deduced and substituted, 
if desired, to provide means of calculating the other thermal 
capacities , but that properly belongs in the discussion of 
perfect gases and will be considered in the proper place. 

For a different substance — for example, superheated steam 
— it will appear that the ratio of the specific heats is not a con- 
stant, and, further, the form of the characteristic equation is 
different. The values of the partial differential coefficients 
must of course be found for each special case, and the use to be 
made of the relations of the thermal capacities will depend 
on circumstances. 



CHAPTER II. 
FIRST LAW OF THERMODYNAMICS. 

The formal statement of the first law of thermodynamics is : 

Heat and mechanical energy are mutually convertible, and 
heat requires for its production and produces by its disappearance 
a definite number of units of work for each thermal unit. 

This law, which may be considered to be the second gen- 
eral principle of thermodynamics, is the statement of a well- 
determined physical fact. It is a special statement of the 
general law of the conservation of energy, i. e., that energy 
may be transformed from one form to another, but can neither 
be created nor destroyed. It should be stated, however, that 
the general law of conservation of energy, though universally 
accepted, has not been proved by direct experiment in all 
cases ; there may be cases that are not susceptible of so direct 
a proof as we have for the transformation of heat into work. 

The best determinations of the mechanical equivalent of 
heat were made by Rowland, whose work will be considered 
in detail in connection with the properties of steam and water. 
From his work it appears that 778 foot-pounds of work are 
required to raise one pound of water from 62 to 63 Fahren- 
heit ; this value of the mechanical equivalent of heat is now 
commonly accepted by engineers, and is verified by the latest 
determinations by Joule and other experimenters. 

The values of the mechanical equivalent of heat for the 
English system and for the metric system are : 

1 B. T. U. = 778 foot-pounds. 

1 calorie = 426.9 metre-kilograms. 

15 



1 6 THERMODYNAMICS OF THE STEAM-ENGINE. 

This physical constant is commonly represented by the letter 
J\ the reciprocal is represented by A. 

In older works on thermodynamics the values of J are com- 
monly quoted as 772 for the English system and 424 for the 
metric system. The error of these values is about one per cent. 

Effects of the Transfer of Heat. — Let a quantity of 
any substance of which the weight is one unit — i. e., one pound 
or one kilogram — receive a quantity of heat dQ. It will, in 
general, experience three changes, each requiring an expendi- 
ture of energy. They are : (1) The temperature will'be raised, 
and, according to the theory that sensible heat is due to the 
vibrations of the particles of the body, the kinetic energy will be 
increased. Let dS represent this change of sensible heat or 
vibration work expressed in units of work. (2) The mean 
positions of the particles will be changed ; in general the body 
will expand. Let dl represent the units of work required 
for this change of internal potential energy, or work of disgre- 
gation. (3) The expansion indicated in (2) is generally against 
an external pressure, and to overcome the same — that is, for 
the change in external potential energy — there will be required 
the work dW. 

If during the transmission no heat is lost, and if no heat is 
transformed into other forms of energy, such as sound, elec- 
tricity, etc., then the first law of thermodynamics gives 

dQ = A(dS+df+dW) (22) 

It is to be understood that any or all of the terms of the 
equation may become zero or may be negative. If all the 
terms become negative heat is withdrawn instead of added, 
and dQ is negative. It is not easy to distinguish between the 
vibration work and the disgregation work, and for many pur- 
poses it is unnecessary ; consequently they are treated together 
under the name of intrinsic energy, and we have 

dQ = A(dS + dI+dW)=A(dE + dW). . (23) 



FIRST LAW OF THERMODYNAMICS. \*J 

The inner work, or intrinsic energy, depends on the state of 
the body, and not at all on the manner by which it arrived at 
that state ; just as the total energy of a falling body, with refer- 
ence to a given plane consisting of kinetic energy and potential 
energy, depends on the velocity of the body and the height 
above the plane, and not on the previous history of the body. 

The external work is assumed to be done by a fluid 
pressure; consequently 

dW — pdv, (24) 

W = fj x *P dv > • • ( 2 ?) 

where v^ and z/, are the final and initial volumes. 

In order to find the value of the integral v in equation (25) 
it is necessary to know the manner in which the pressure varies 
with the volume. Since the pressure may vary in different 
ways, the external work cannot be determined from the initial 
and final states of the body; consequently the heat required 
to effect a change from one state to another depends on the 
manner in which the change is effected. 

Assuming the law of the variation of the pressure and 
volume to be known, we may integrate thus: 

Q = a[e, -£,+ £*pdv) (26) 

In order to determine E for any state of a body it would 
be necessary to deprive it entirely of vibration and disgregation 
energy, which would of course involve reducing it to a state 
of absolute cold ; consequently the direct determination is 
impossible. However, in all our work the substances operated 
on are changed from one state to another, and in each state 
the intrinsic energy depends on the state only; consequently 
the change of intrinsic energy may be determined from the 
initial and final states only, without knowing the manner 
of change from one to the other. 



i8 



THERMODYNAMICS OF THE STEAM-ENGINE. 



All succeeding equations will be arranged to involve differ- 
ences of energy only, and the hypothesis involved in a separa- 
tion into vibration and disgregation work avoided. 

Thermal Lines. — The external work can be determined 
only when the relations of / and v are known, or, in general, 
when the characteristic equation is known. It has already been 
shown that in such case the equation may be represented by a 
geometrical surface, on which so-called thermal lines can be 
drawn representing the properties of the substance under con- 
sideration. These lines are commonly projected on the (/, v) 
plane. It is convenient in many cases to find the relation of/ 
and v under a given condition and represent it by a curve drawn 
directly on the (/, v) plane. 

Lines of Equal Pressure. — The change of 
condition takes place at constant pressure, and 
consists of a change of volume, as represented in 
Fig. 2. The tracing-point moves from a 1 to a„ 
, and the volume changes from v y to v 2 . The 
work done is represented by the rectangular area 



Fig. 2. 
under #/z 2 , or by 



W 






During the change the temperature may or may not change ; 
the diagram shows nothing concerning it. 

Lines of Equal Volume. — The pressure in- 
creases at constant volume, and the tracing-point 
moves from a x to # 2 . The temperature usually 
increases meanwhile. Since dv is zero, 



W 



■■ I pdv = o. 



-a« 



Fig. 3. 



Isothermal Lines or Lines, of Equal Temperature. — 

The temperature remains constant, and a line is drawn, usually 
convex, toward the axis OV. The pressure of a mixture of a 




FIRST LAW OF THERMODYNAMICS. 1 9 

liquid and its vapors is constant for a given temperature ; con- 
sequently the isothermal for such a mixture is a line of equal 
pressure, represented by Fig. 2. The iso- 
thermal of a perfect gas, on the other hand, is 
an equilateral hyperbola, as appears from the 
law of Boyle, which may be written 

pv = C (27) Fig. 4. 

Isodynamic or Isoenergic Lines are lines representing 
changes during which the intrinsic energy remains constant. 
Consequently all the heat received is transformed into external 
work. It will be seen later that the isodynamic and isothermal 
lines for a gas are the same. 

Adiabatic Lines. — A very important problem in thermo- 
dynamics is to determine the behavior of a substance when 
a change of condition takes place in a non-conducting vessel. 
During the change — for example, an increase of volume or ex- 
pansion — some of the heat in the substance may be changed 
into work ; but no heat is transferred to or from the sub- 
stance through the walls of the containing vessel. Such 
changes are called adiabatic or isoentropic changes. 

Very rapid changes of dry air in the cylinder of an air- 
compressor or a compressed-air engine are very nearly adi- 
abatic. Adiabatic changes never occur in the cylinder of a 
steam-engine on account of the rapidity with which steam is 
condensed on or vaporized from the cast-iron walls of the 
cylinder. 

Since there is no transmission of heat to (or from) the 
working substance, equation (26) becomes 



Q = A{E-E X - J pdv), 

V 




20 THERMODYNAMICS OF THE STEAM-ENGINE. 

that is, the external work is done wholly at the expense of 
the intrinsic energy of the working substance, as must be the 
case in conformity with the assumption of an adiabatic 
change. 

Relation of Adiabatic and Isothermal Lines. — An adia- 
batic line drawn on the (p t v) plane is steeper than an isother- 
mal line at the point of intersection. This is easily shown 
for a substance that expands with a rise of 
temperature. Thus let ab and cd (Fig. 5) 
represent an adiabatic and an isothermal line 
crossing at p. The substance when in the 
d condition represented by the point / has a 
certain volume, pressure, and temperature, 
The isothermal change represented by pd, is 




Fig. 5- at constant temperature. On the other 

hand, the adiabatic change represented by pb y is accompanied 
by a loss of intrinsic energy ; but the intrinsic energy is the 
sum of the vibration energy and the disgregation energy, 
and, in general, a loss of intrinsic energy means a diminution 
in both vibration and disgregation energy. Now the vibra- 
tion energy is represented by the temperature, and the tem- 
perature will fall when the vibration energy decreases. Con- 
sequently the temperature at b is less than the temperature at 
p, and therefore is less than the temperature at d. But b 
and d are at the same pressure, and consequently the volume 
at b is less than the volume at d\ that is, the adiabatic line is 
the steeper. 

Graphical Representations of Change of Intrinsic 
Energy. — Professor Rankine first used a graphical method of 
representing a change of intrinsic energy, employing adiabatic 
lines only, as follows: 

Suppose that a substance is originally in the state A (Fig. 
6), and that it expands adiabatically ; then the external work 
is done at the expense of the intrinsic energy , hence if the 
expansion has proceeded to A 1 the area AA x a x a y which repre- 



p 












B 


fv 




o 




N 


^ 






Z> a 


a, 






Fig 


6. 





FIRST LAW OF THERMODYNAMICS. 21 

sents the external work, also represents the change of intrinsic 
energy. Suppose that the expansion were to continue indefi- 
nitely : then the adiabatic will approach the 
axis OV indefinitely, and the area repre- 
senting the work will be included between 
the curve Aa produced indefinitely, the 
ordinate Aa, and the axis OV; this area will 
represent all the work that can be obtained 
by the expansion of the substance; and if it 
be admitted that during the expansion all the intrinsic energy is 
transformed into work, so that at the end the intrinsic energy 
is zero, it represents also the intrinsic energy. In cases for 
which the equation of the adiabatic can be found it is easy to 
show that 

E x = / pdv 

is a finite quantity ; and' in any case, if we admit an absolute zero 
of temperature, it is evident that the intrinsic energy cannot 
be infinite. On the other hand, if an isothermal curve were 
treated in the same way the area would be infinite, since heat 
would be continually added during the expansion. 

Now suppose the body to pass from the condition repre- 
sented by A to that represented by B, by any path whatever — 
that is, by any succession of changes whatever — for example, 
that represented by the irregular curve AB. The intrinsic 
energy in the state B is represented by the area VbB/3. The 
change of intrinsic energy is represented by the area /SBbaAa, 
and this area does not depend on the form of the curve AB. 
This graphical process is only another way of saying that the 
intrinsic energy depends on the state of the substance only, 
and that change of intrinsic energy depends on the final and 
initial states only. 

Another way of representing change of intrinsic energy by 



22 



THERMODYNAMICS OF THE STEAM-ENGINE. 




Fig. 7. 



aid of isodynamic lines avoids an infinite diagram. Suppose 

the change of state to be represented by the 
curve AB (Fig. 7). Draw an isodynamic line 
A £7 through the point A, and an adiabatic 
line BC through B, intersecting at C. Then 
the area^ABba represents the external work, 
and the area bBCc represents the change of 
intrinsic energy ; for if the body be allowed 
to expand adiabatically till the intrinsic energy is reduced to 
its original amount at the condition represented by A the 
external work bBCc will be done at the expense of the intrinsic 
energy. And further, since the intrinsic energy is constant for 
all points on the isodynamic line through A, and in like man- 
ner is constant for points on the line through B, there will be 
the same change of intrinsic energy in passing from a condition 
represented by any point of the line through A to any point 
of the line through B; consequently if through any point, 
as D of the upper line, an adiabatic DE be drawn the area 
dDEe will be equal to bBCc, and will equally represent the 
change of intrinsic energy from the point A to the point B. 

Entropy. — If a body have its condition represented by the 
point e of the isothermal aa x (Fig. 8) it will have a definite 
temperature, which will be the same so long as 
its condition is represented by some point on 
aa„ as, for example, e,, though the volume and 
pressure will meanwhile have varied. Should 
the temperature change, the condition will 
be represented by some point, as f> on Fig. 8. 

another isothermal bb y . There will evidently be the same 
change of temperature in passing from e to f as from e\ 
to/", ; that the changes of volume and pressure, external work, 
and intrinsic energy are different does not affect the statement 
concerning the temperature. In like manner it is indifferent 
how or at what part of the diagram the transfer from bb 1 to cc x 
is accomplished ; the same change of temperature must occur. 
In the same way isoenergic changes will be represented by 





FIRST LA W OF T HER MOD YNAMICS. 2$ 

the motion of a point along a curve of constant energy ; and 
there will be a definite change of energy in passing from a 
curve of constant energy to the next curve of a system of 
isoenergic curves. 

Conversely, if we have any system of curves it is reason- 
able to suppose that there must be a constant change of some 
sort in passing from one such curve to the next of the same 
system. A series of adiabatic curves as represented by Fig. 
9, is such a system of curves, and we may con- 
sider that there is the same change in passing 
from e to /"as in passing from e l to f v e being 
any point on the curve aa x and f being any 
point on the next curve bb x . It will appear 
in our future work that a definite form may Fig. 9. 

be assigned to the function representing the change that 
occurs in passing from one adiabatic line of a given sub- 
stance to another adiabatic line of the same substance, and 
that a numerical calculation may be made representing that 
change. It will further appear that the form of the function 
and the corresponding numerical calculation will depend on 
the nature of the substance, and will be different for different 
substances. For example, the form of the function for steam 
is radically different from the form for air. The form of the 
function is consequently a property of the substance, just as 
are specific volume and intrinsic energy. The fact that the 
form of the function cannot be intelligently explained now, 
and that the nature is different from other properties thus far 
discussed, is no reason why we should not provide for the 
function, give it a name, and include it in our equations. In 
the proper place the form and use of the function will be ex- 
plained. 

The name given to this function or property which re- 
mains constant during an adiabatic change is entropy. It is 
commonly represented by <p. 

In the process of establishing an absolute scale of temper- 
ature we shall show how a system of isothermal lines can be 



24 THERMODYNAMICS OF THE STEAM-ENGINE. 

drawn at intervals of one degree of temperature. At the 
same time, and by a similar method, a system of adiabatics 
will be drawn, each one unit of entropy from the next. Both 
systems of lines will be made to depend on the foot-pound. 
It may be suggested further that thus far no adequate idea 
of the nature of temperature has been obtained, the assertion 
that temperature is in some way connected with the vibration 
energy of a body being too indefinite for this purpose. 
Though an accepted dictum of science, it remains to a certain 
extent speculative. Again, for the engineer it is more im- 
portant to be able to calculate changes and effects than to be 
able to give a philosophical account of their real nature. 



CHAPTER III. 
SECOND LAW OF THERMODYNAMICS. 

Heat-engines are engines by which heat is transformed 
into work. All actual engines used as motors go through con- 
tinuous cycles of operations, which periodically return things 
to the original conditions. All heat-engines are similar in that 
they receive heat from some source, transform part of it into 
work, and deliver the remainder (minus certain losses) to a 
refrigerator. 

The source and refrigerator of a condensing steam-engine 
are the furnace and the condenser. The boiler is properly con- 
sidered as a part of the engine, and receives heat from the 
source. 

Carnot's Engine. — It is convenient to discuss a simple 
ideal engine, first described by Carnot. 

Let P of Fig. 10 represent a cylinder with non-conducting 
walls, in which is fitted a piston, also of non-conducting mate- 
rial, and moving without friction ; on 
the other hand, the bottom of the 
cylinder is supposed to be of a material 
that is a perfect conductor. There is 



a non-conducting stand C on which a | u c u' | b 
the cylinder can be placed while Fig. io. 

adiabatic changes take place. The source of heat A at a 
temperature t is supposed to be so maintained that in 
operations during which the cylinder is placed on it, and 
draws heat from it, the temperature is unchanged. The 
refrigerator B at the temperature /, in like manner can with- 

25 




26 THERMODYNAMICS OF THE STEAM-ENGINE. 

draw heat from the cylinder, when it is placed on it, at a con- 
stant temperature. 

Let there be a unit of weight (for example, one pound) of 
a certain substance in the cylinder at the temperature t of the 
source of heat. Place the cylinder on the source of heat A 
(Fig. 10), and let the substance expand at the constant tem- 
perature T, receiving heat from the source A. 

If the first condition of the substance be 
represented by A (Fig. n), then the second 
will be represented by B, and AB will be an 
isothermal. If E a and E b are the intrinsic 
energies at A and B, and if W ab , represented 
Fig. ii. by the area aABb, be the external work, the 

heat received from A will be 

Q = A(E b -E a + W«). 9 

Now place the cylinder on the stand C (Fig. 10), and let 
the substance expand adiabatically until the temperature is 
reduced to T xi that of the refrigerator, the change being rep- 
resented by the adiabatic BC (Fig. n). If E c is the intrinsic 
energy at C, then, since no heat passes into or out of the 
cylinder, 

o = A(E c -E b +W bc ), 

where W bc is the external work represented by the area bBCc. 
Place the cylinder on the refrigerator B, and compress the 
substance till it passes through the change represented by CD, 
yielding heat to the refrigerator so that the temperature re- 
mains constant. If E d is the intrinsic energy at D, then 

- 0, = A{E d -E c - W cd ) 

is the heat yielded to the refrigerator, and W cd , represented by 
the area cCDd, is the external work, which has a minus sign, 
since it is done on the substances. 






SECOND LAW OF THERMODYNAMICS. 2? 

The point D is determined by drawing an adiabatic from 
A to intersect an isothermal through C. The process is com- 
pleted by compressing the substance while the cylinder is on 
the stand C (Fig. 10) till the temperature rises to T, the 
change being represented by the adiabatic DA. Since there 
is no transfer of heat, 

o = A(E. -E d - W da ). 

Adding together the several equations, member to mem- 
ber, 

Q- a = A{W ab + W bc - W cd - W da ); 

or, if Wbe the resulting work represented by the area ABCD> 
then 

Q-Q, = AW; 

that is, the difference between the heat received and the heat 
delivered to the refrigerator is the heat transformed into work. 

A Reversible Engine is one that may run either in the 
usual manner, transforming heat into work, or reversed, 
describing the same cycle in the opposite direction, and trans- 
forming work into heat. 

A Reversible Cycle is the cycle of a reversible engine. 

Carnot's engine is reversible, the reversed cycle being 
ADCBA (Fig. n), during which work is done by the engine 
on the working substance. The engine then draws from the 
refrigerator a certain quantity of heat, it transforms a certain 
quantity of work into heat, and delivers the sum of both to 
the source of heat. 

No actual heat-engine is reversible in the sense just stated, 
for when the order of operations can be reversed, changing 
the engine from a motor into a pump or compressor, the re- 
versed cycle differs from the direct cycle. For example, the 
valve-gear of a locomotive may be reversed while the train is 
running, and then the cylinders will draw gases from the 
smoke-box, compress them, and force them into the boiler. 



28 



THERMODYNAMICS OF THE STEAM-ENGINE. 



The locomotive as ordinarily built is seldom reversed in this 
way, as the hot gases from the smoke-box injure the surfaces 
of the valves and cylinders. Some locomotives have been 
arranged so that the exhaust-nozzles can be shut off and 
steam and water supplied to the exhaust-pipe, thus avoiding 
the damage from hot gases when the engine is reversed 
in this way. Such an engine may then have a reversed cycle, 
drawing steam into the cylinders, compressing and forcing it 
into the boiler; but in any case the reversed cycle differs 
from the direct cycle, and the engine is not properly a revers- 
ible engine. 

A Closed Cycle is any cycle in which the final state is the 

same as the initial state. Fig. 12 represents 

such a cycle made up of four curves of any 

nature whatever. If the four curves are of two 

v species only, as in the diagram representing 

Fig. 12. the cycle of Carnot's engine, the cycle is said 

to be simple. In general, we shall have for a cycle like that 

of Fig. 12 




Qa b + Q bc - Q cd - Q, 



da 



= A{W ab + W hc - W cd - W da ). 



p 


A 


B 






c 


V 


^ 






u 






V 



Fig. 13. 



A closed curve of any form may be consid- 
ered to be the general form of a closed cycle, 
as that in Fig. 13. For such a cycle we have 

/ dQ = A I dW, which is one more way of 

stating the first law of thermodynamics. 

It may make this last clearer to con- 
sider the cycle of Fig. 14, composed of the 
isothermals AB, CD, and EG, and the 
adiabatics BC, DE, and GA. The cycle 
may be divided by drawing the curve 
Fig. 14. through from C to F. It is indifferent 

whether the path followed be ABCDEGA or ABCFCDEGA 
or, again, ABC EG A + CDEFC. 





SECOND LA W OF THERMOD YNAMICS. 2$ 

Again, an irregular figure may be imagined to be cut into 
elementary areas by isothermals and adia- 
batic lines, as in Fig. 15. The summation 
of the areas will give the entire area, and 
the summation of the works represented 
by these will give the entire work repre- 
sented by the entire area. 

The Efficiency of an engine is the Fig. 15. 

ratio of the heat changed into work to the entire heat applied ; 
so that if it be represented by 7;, 

AW Q-Q 
^-Q-^—Q- (29) 

for the heat Q rejected to the refrigerator is what is left after 
A W thermal units have been changed into work. 

Carnot's Principle. — It was first pointed out by Carnot 
that the efficiency of a reversible engine does not depend on 
the nature of the working substance, but that it depends on 
the temperatures of the source of heat and the refrigerator. 

Let us see what would be the consequence if this princi- 
ple were not true. Suppose there are two reversible engines 
R and A, each taking Q thermal units per second from the 
source of heat, of which A is the more efficient, so that 



is larger than 



A W a Q - q: 

~Q" = -Q- (28) 

AW r _Q~Q r \ 



Q Q 



(29) 



this can happen only because QJ is less than Q r \ for Q is as- 
sumed to be the same for each engine. Let the engine R be 
reversed and coupled to A, which can run it and still have left 
the useful work W a — W r . This useful work cannot come 
from the source of heat, for the engine R when reversed gives 



30 THERMODYNAMICS OF THE STEAM-ENGINE. 

to the source Q thermal units per second, and A takes the 
same amount in the same time. It must be assumed to come 
from the refrigerator, which received Qd thermal units per 
second, and gives up QJ thermal units per second, so that it 
loses 

Qr'-Q.' =A(W a - W r ) 

thermal units per second. This equation may be derived 
from equations (28) and (29) by subtraction. 

Now it cannot be proved by direct experiment that such 
an action as that just described is impossible. Again, the first 
law of thermodynamics is scrupulously regarded, and there is 
no contradiction or formal absurdity of statement. And yet 
when the consequences of the negation of Carnot's principles 
are clearly set forth they are naturally rejected as improbable, 
if not impossible. The justification of the principle is found 
in the fact that theoretical deductions from it are confirmed 
by experiments. 

Second Law of Thermodynamics. — The formal state- 
ment of Carnot's principle is known as the second law of ther- 
modynamics. Various forms are given by different investiga- 
tors, none of which are entirely satisfactory, for the conception 
is not simple, as is that of the first law. 

The following are some of the statements of the second 
law : 

(1) All reversible engines working between the same source 
of heat and refrigerator have the same efficiency. 

(2) The efficiency of a reversible engine is independent of 
the working substance. 

(3) A self-acting machine cannot convey heat from one body 
to another at a higher temperature. 

The second law is the third general principle of thermo- 
dynamics ; it differs from each of the others and is independ- 
ent of them. Summing up briefly, the first general principle 
is a pure assumption that thermodynamic equations may con- 






SECOND LAW OF THERMODYNAMICS. 3 1 

tain only two independent variables; the second is the state- 
ment of an experimental fact ; the third is a choice of one of 
the two propositions of a dilemma. The first and third are 
justified by the results of the application of the theory of 
thermodynamics. 

Carnot's Function. — Carnot's principle asserts that the 
efficiency of a reversible engine is independent of the nature 
of the working substance ; consequently the expression for the 
efficiency will not include such properties of the working sub- 
stance as specific volume and specific pressure. But the prin- 
ciple asserts also that the efficiency depends on the tempera- 
tures of the source of heat and the refrigerator, which indeed 
are the only properties of the source and refrigerator that can 
affect the working of the engine. 

We may then represent the efficiency as a function of the 
temperatures of the source of heat and the refrigerator, or, what 
amounts to the same thing, as a function of the superior tem- 
perature and the difference of the temperatures, and may 
write 

AW Q-Q! u . 4 - . N 

where Q is the heat received, Q the heat rejected, and t and 
t' are the temperatures of the source of heat and of the refrig- 
erator on any scale whatsoever, absolute or relative. 

If the temperature of the refrigerator approaches near that 
of the source of heat Q — Q and t — t' become A Q and At, 
and at the limit dQ and dt, so that 

d -^ = F(t,df) (31) 

But dt is itself a function of t, so that at the limit the effi- 
ciency depends on t only. It is convenient to express the 
equation in the form 

d Q=/(t)dt. ...... (32) 



32 THERMODYNAMICS OF THE STEAM-ENGINE. 

An algebraic reduction from equation (31) to (32) may be 
made as follows : divide, and multiply by dt, so that 

Q at 

f(t) is called Carnot's function and is represented by >u, 
so that the efficiency may be represented by 

dQ dt 

-Q = V™' 

The form of the function will depend on the scale of temper- 
ature selected and will vary from one scale to another scale. 
The logical method appears to be to choose some scale of tem- 
perature and deduce the form of Carnot's function correspond- 
ing. The easier way is to define or establish an arbitrary 
scale, independent of any special thermometer or material, 
and find the difference between that scale and the scale of a 
thermometer such as the air thermometer or a mercurial 
thermometer. 

Absolute Scale of Temperature. — The simplest form that 

can be assigned to Carnot's function is -- , where T is the 

absolute temperature on the arbitrary scale corresponding to 
that form of Carnot's function. The scale of temperature so 
determined is the absolute scale referred to on page 5, and 
depends only on the fundamental units the foot and the 
pound, or, what amounts to the same thing, on the foot- 
pound. The most ready way of showing this is by Thom- 
son's graphical method. 

Thomson's Graphical Method. — The method just given 
of arriving at an absolute scale independent of any substance 
was first given by Lord Kelvin, who further explained it by 
the following graphical construction : 

In Fig. 16 let ak and bi be two adiabatic lines, and let 




Fig. i 6. 



SECOND LA W OF THERMOD YNAMICS. 33 

the substance have its condition represented by the point a. 
Through a and d draw iso- 
thermal lines; then the dia- 
gram abed represents thecycle 
of a simple reversible engine. 
Draw the isothermal line fe, 
so that the area dcef shall be 
equal to abcd\ then the dia- 
gram dcef represents the cycle 
of a reversible engine, doing 
the same amount of work per stroke as that engine whose cycle 
is represented by abed; and the difference between the heat 
drawn from the source and delivered to the refrigerater — i. e., 
the heat transformed into work — is the same. The refrigerator 
of the first engine might serve for the source of heat for the 
second. 

Suppose that a series of equal areas are cut off by isother- 
mal lines, as fegh, hgik, etc., and suppose there are a series of 
reversible engines corresponding : then there will be a series of 
sources of heat of determinate temperatures, which may be 
chosen to establish a thermometric scale. In order to have 
the scale correspond with those of ordinary thermometers one 
of the sources of heat must be at the temperature of boiling 
water, and one at that of melting ice ; and for the centi- 
grade scale there will be one hundred, and for the Fahrenheit 
scale one hundred and eighty, such cycles, with the appropriate 
sources of heat, between boiling-point and freezing-point. To 
establish the absolute zero of the scale the series must be im- 
agined to be continued till the area included between an iso- 
thermal and the two adiabatics, continued indefinitely, shall 
not be greater than one of the equal areas. 

This conception of the absolute zero may be made clearer 
by taking wide intervals of temperature, as on Fig. 1 7, where the 
cycle abed is assumed to extend between the isothermals of o° 
and ioo° C. ; that is, from freezing-point to boiling-point. The 
next cycle, cdef, extends to — ioo° C.,and the third cycle, efgh> 



34 



THERMODYNAMICS OE THE STEAM-ENGINE. 




extends to — 200 ° C. The remaining area, which is of infinite 

length and extremely attenuated, 
is bounded by the isothermal gh 
and the two adiabatics ha and g/3. 
The diagram of course cannot be 
completed, and consequently the 
area cannot be measured ; but 
when the equations to the isother- 
mal and the adiabatics are known 
it can be computed. So com- 
puted, the area is found to be 

72 7 

-Z-t- of one of the three equal 
ioo 

areas abed, cdfe, and efhg. The 
absolute zero is consequently 
"v 2 73°-7 C. below freezing-point. 
Fig. 17. Further discussion of the absolute 

scale will be deferred till a comparison is made with the air- 
thermometer. 

Scale of Entropy. — A similar treatment may be given 
to the scale of entropy. Thus in Fig. 18 let the iso- 
thermals ab and cd be ex- 
tended indefinitely, and let 
a series of adiabatics be 
drawn, cutting off equal areas 
abed, blmc, and Inom, etc. ; 
we shall then have a series of 
intervals of entropy depend- 
ing on the foot-pound. To 
make the scale of entropy 
definite we will assume that 
the isothermals an and do 
are one degree apart, and that the initial cycle abed is so drawn 
that it represents the change of one thermal unit into work ; 
for the English system the area abed represents 778 foot- 
pounds. 




Fig. 18. 



SECOND LAW OF THERMODYNAMICS. 35 

Now the area beneath an isothermal line extending indefi- 
nitely is infinite, for heat is continually added to the working 
substance to keep up the temperature, and there is no limit to 
the amount of work that can be done by expansion. In this 
regard the isothermal is radically different from the adiabatic, 
for an infinite adiabatic expansion is supposed to change all 
the intrinsic energy of the working substance into external 
work ; and as the intrinsic energy is finite, so also must be the 
area representing external work. We may conclude that the 
area bounded by the adiabatic ad, and the two isothermals 
an and do, is infinite, and that there is no limit to the number 
of equal areas that can be cut off by equally spaced adiabatics. 

It will be noted that on the diagram the adiabatic no is 
higher than the adiabatic ad, so that the entropy increases 
from a towards n. The area of the strip between the adia- 
batic be and the isothermals ba, and ed, both extended indefi- 
nitely towards the left, is infinite; consequently there is no 
absolute zero of entropy, and we shall therefore be able to 
calculate differences of entropy only. The area of the typical 
cycle chosen for measuring intervals of entropy is large (equal 
to 778 foot-pounds) ; consequently the numbers expressing 
changes of entropy will be found to be small. 

Efficiency of Reversible Engines — The general differ- 
ential equation for the efficiency of a reversible engine is 

dO 

' =f{f)dt = }xdt, 



Q 



or, making Carnot's function equal to -~,, 

dQ _ dt_ 
~Q~~f' 

Integrating between limits, 

Q! T' 

logc Q =logt Y* 



36 THERMODYNAMICS OF THE STEAM-ENGINE. 

,. ,^-i^r (33 ) 



The absolute scale of temperature consequently depends 
only on the efficiency of a reversible engine, and since the ef- 
ficiency of such an engine is independent of the properties of 
any substance, so also is the absolute scale. The efficiency is 
expressed in thermal units, which are equivalent to the proper 
number of mechanical units (foot-pounds) ; consequently the 
absolute scale of temperature may be made to depend directly 
on the foot-pound — that is, on the fundamental units the foot 
and the pound. This discussion is only another way of view- 
ing the ideas discussed by Thomson's graphical method. 
Graphical Representation of Efficiency. — Let Fig. 19 
represent the cycle of a reversible heat- 
engine. For convenience it is supposed 
there are four degrees of temperature 
from the isothermal AB to the isother- 
mal DC, and that there are three inter- 
vals or units of entropy between the 
adiabatics AD and BC. First it will be 
— shown that all the small areas info which 




the cycle is divided by drawing the inter- 
vening adiabatics and isothermals are equal. Thus we have 
to begin with a = b and a = c by construction. But engines 
working on the cycles a and b have the same efficiency and 
reject the same amounts of heat. These heats rejected are 
equal to the heats supplied to engines working on the cycles 
c and d, which consequently take in the same amounts of heat. 
But these engines work between the same limits of tempera- 
ture and have the same efficiency, and consequently change 
the same amount of heat into work. Therefore the areas c 
and d are equal. In like manner all the small areas are equal, 



SECOND LAW OF THERMODYNAMICS. 37 

and each represents one thermal unit, or 778 foot-pounds of 
work. 

It is evident that the heat changed into work is repre- 
sented by 

(T-T'){4>'-4>), (34) 

and, further, that the same expression would be obtained for a 
similar diagram, whatever number of degrees there might be 
between the isothermals, or intervals of entropy between the 
adiabatics, and that it is not invalidated by using fractions of 
degrees and fractions of units of entropy. It is consequently 
the general expression for the heat changed into work by an 
engine having a reversible cycle. 

It is clear that the work done on such a cycle increases as 
the lower temperature T' decreases, and that it is a maximum 
when T' becomes zero, for which condition all of the heat 
applied is changed into work. Therefore the heat applied is 
represented by 

Q = T(<p' - 4>), (35) 

and the efficiency of the engine working on the cycle repre- 
sented by Fig. 19 is 

AW _ Q- Q (T- T)(cf> f -cp) T - T' 
Q ~ Q ~~ T(cP'-cp) ~ T ' 

as found by equation (33). The deduction of this equation 
by integration is more simple and direct, but the graphical 
method is interesting and may give the student additional 
light on this subject. 

Temperature — Entropy Diagram. — Thermal diagrams 
are commonly drawn with pressure and volume for the coordi- 
nates, but for some purposes it is convenient to use temperature 
and entropy. Thus Carnot's cycle, when temperature and 



38 



THERMODYNAMICS OF THE STEAM-ENGINE. 



<p 




q>> 




1 
T 

• 





























<P 



Fig. 20. 



entropy are used for coordinates, becomes a rectangle, as 
shown by Fig. 20. As in Fig. 19, four de- 
grees of temperature and three intervals of 
t entropy have been chosen, and the diagram is 
subdivided into twelve equal areas, which in 
- Fig. 20 are evidently equal, as they are all 
rectangles. 
Expression for Entropy. — Resuming the expression 

(35) for the heat supplied 

along the isothermal line AB 

(Fig. 19), it appears that the 

amount of heat supplied de- 
pends on the temperature T 

and on the difference of en- 
tropy 0' — 0. The amount 

of heat will decrease as the 

interval of entropy decreases, 

and will approach dQ when 

<p' — <p approaches d(p y as 

shown by Fig. 21, so that 




Fig. 21. 



dQ = Td4>, 



or 



d<p = ~, 



(36) 



or, integrating between limits, 



0'-0 = 




(37) 



Equation (37) may be used for calculating the change 
of entropy during any reversible operation, provided that the 
heat added may be expressed as a function of the temperature. 



SECOND LAW OF THERMODYNAMICS, 39 

For example, we may calculate the change of entropy 
from A to B (Fig. 19) as follows: 

The temperature during the isothermal expansion from A 
to B is constant, therefore the heat added is due entirely to 
the latent heat of expansion, and we have from equation (5) 

dQ = Idv. 

For a perfect gas the latent heat of expansion is given 
by equation (66), page 63, so that 

P 
dQ = (cp — c v )J—dv. 

R 

But R = — - ; consequently 

ja, dQ . dv 

d(p = -=- — {cp — c v )—. 
1 v 

.-. <t>' -<P= (*,-*,) log, -\ 

Suppose that v b is 8 cubic feet, and v a is 4 cubic feet, then 

o 

0' - = (0.2375 — O.I69O) log, — , 

4 
when the values of c P and c v for air are used, giving finally 

— 0' = O.O685 X O.693I = O.O475 

for the increase of entropy corresponding to the isothermal 
expansion from A to B. 

Application to a Reversible Cycle. — A very important 
result is obtained by the application of equation (37) to the 
calculation of entropy during a reversible cycle. In the first 
place, it is clear that the entropy of a substance having its 
condition represented by the point a (Fig. 22), depends on the 



40 



THERMODYNAMICS OF THE STEAM-ENGINE. 




adiabatic line drawn through it ; in other words, the entropy 

depends only on the condition of 
the substance. In this regard 
entropy is like intrinsic energy 
and differs from external work. 
Suppose now that the substance 
V is made to pass through a cycle 
Fig. 22. of operations, represented by the 

point a tracing the diagram on Fig. 22 : it is clear that the 
entropy will be the same at the end of the cycle as at the 
beginning, for the tracing-point will then be on the original 
adiabatic line. As the tracing-point moves toward the right 
from adiabatic to adiabatic the entropy increases, and as 
it moves to the left the entropy decreases, the algebraic sum 
of changes of entropy being zero for the entire cycle. This 
conclusion holds whether the cycle is reversible or non- 
reversible. 

If the cycle is reversible, then equation (37) may be used for 
calculating the several changes of entropy, and for calculating 
the change for the entire cycle, giving for the cycle 




= o. 



(38) 



This is a very important conclusion from the second law of 
thermodynamics, and is considered to represent that law. The 
second law is frequently applied by using this equation in con- 
nection with a general equation or a characteristic equation, 
in a manner to be explained later. 

Though the discussion just given is simple and complete, 
there is some advantage in showing that equation (38) holds 
for certain simple and complex reversible cycles. 

Thus for Carnot's cycle, represented by Fig. 19, the 
increase of entropy during isothermal expansion is 



0- <t> = i 



dQ 
T 




SECOND LAW OF THERMODYNAMICS. 



41 



because the temperature is constant. In like manner the 
decrease during isothermal expansion is 







0' = 2, 



so that the change of entropy for the cycle is 



Q_ 
T 



QL 



But from the efficiency of the cycle we have 



Q-Q! T-T 
Q ~ T ' 



QL 
Q 










Fig. 23. 



A complex cycle like that represented by Fig. 23 may be 
broken up into two simple cycles ABFG 
and CDFE, for each of which individually 
the same result will be obtained — that is, 
the increase of entropy from A to B is 
equal to the decrease from F to G, and 
the increase from C to D is equal to the 
decrease from E to F, so that the sum- 
mation of changes for the entire cycle gives zero. 

Fig. 24 represents the simplified ideal diagram of a hot-air 
engine, in which by the aid of a regenerator the adiabatic 

lines of Carnot's cycle are re- 
placed by vertical lines without 
affecting the reversibility or the 
efficiency of the cycle. We may 
replace the actual diagram by a 
series of simple cycles made up of 
isothermals and adiabatics, so 
drawn that the perimeter of the 
_y complex cycle includes the same 




Fig. 24. area and corresponds approxi- 

mately with that of the actual diagram. The summation of 



42 THERMODYNAMICS OF THE STEAM-ENGINE. 

the change of entropy for the complex cycle is clearly zero, as 
before. But by drawing the adiabatic lines near enough to- 
gether we may make the perimeter approach that of the actual 
diagram as nearly as we please, and we may therefore con- 
clude that the integration for the changes of entropy for that 
cycle is also zero. 

Non-reversible Cycles.— If a process or a cycle is non- 
reversible, then the change of entropy cannot be calculated by 
equation (37), and equation (38) will not hold. The entropy 
will, indeed, be the same at the end as at the beginning of the 

cycle, but the integration of -^ for the cycle will not give 

zero. On the contrary, it can be shown that the integration 

of -¥- for the entire cycle will give a negative quantity. 

Thus let the non-reversible engine A take the same amount 
of heat per stroke as the reversible engine R which works on 
Carnot's cycle, but let it have a less efficiency, so that 

q-q: , q-q .\ 

—q- < —q- (39) 

where Q/ represents the heat rejected by the engine A. 
Then 

Q-Q,' < Q-Q' = {T- T'M- <t>\ . . (40) 

Suppose now that T' approaches zero and that 0' approaches 
0, then at the limit we shall have 

dQ, <dQ= Td<p, 
or 

dQ ' s j* 

Integrating for the entire cycle, we shall have 
'dQ, 



f 



T 




< o. .-. / ^=-N, . . (41) 



SECOND LA W OF THERMO D YNAMICS. 43 

where N represents a negative quantity. The absolute value 
of iVwill, of course, depend on the efficiency of the non-re- 
versible engine. If the efficiency is small compared with that 
of a reversible engine, then the value of iVwill be large. If 
the efficiency approaches that of a reversible engine, then N 
approaches zero. It is scarcely necessary to point out that N 
cannot be positive, for that would infer that the non-reversi- 
ble engine had a greater efficiency than a reversible engine 
working between the same temperatures. 

Some non-reversible operations, like the flow of gas 
through an orifice, result in the development of kinetic en- 
ergy of motion. In such case the equation representing the 
distribution of energy contains a fourth term K to represent 
the kinetic energy, and equation (22) becomes 

dQ = A(dS + d/+dW-\-dK). . . . (42) 

As before 5 represents vibration work, / represents disgre- 
gation work, and ^represents external work. If the vibration 
and disgregation work cannot be separated, then we may write 

dQ = A(dE + dW '+ dK) (43) 



CHAPTER IV. 
FUDNAMENTAL EQUATIONS. 

Application of the First Law. — The application of the 
first law of thermodynamics consists in uniting an equation 
resulting from that law to some general or some characteristic 
equation. For example, equations (5) and (23) give 

dQ = A(dE + dW) = c v dt + ldv y 

or, replacing dW by pdv, 

A(dE -\-pdv) = c v dt -f- Idv. 

.-. d£= C -jdt + [i -p)dv. . . . (44) 

Now E depends on the state of the body only, and not on 
the method of changing from one condition to another; that 
is, dE is an exact differential, and consequently 

d'E d' l E 



dt dv dv dt ' 
which may be written 

\dt'-» \ =< ^dv't 



dv ) , dt 



FUNDAMENTAL EQUATIONS. 45 

in which the partial differential coefficients are those of the 
equation 

dE = &i dt + if) 4v. 
\dti \dvlt 

Comparing with equation (44), it appears that 

©.-* - ®. ■*&->)■ 

and that consequently 



d lc v \ d i I 



dv\A)t dtxA 



ir 
A 



S),"(s),] = (S'.' • • • <4S) 



Equation (45) represents the relation which must exist be- 
.ween the latent heat of expansion and the specific heat at 
constant volume in consequence of the first law of thermody- 
namics. In order that the relation may be developed for 
some particular substance, such as air, the partial differential 

tion. The use of this and of similar succeeding equations can 
be determined only by the application of the theory of ther- 
modynamics to various substances. 

In a similar manner the first law may be applied to equa- 
tion (6), as follows : 

dQ = A(dE-\-pdv) = c p dt + mdp. 

Substituting the value of dv from the equation 

(dv\ T . , idv\ 



*-©.*+ s> * 



4 6 



THERMODYNAMICS OF THE STEAM-ENGINE. 

A [dE +p( d £) dp + /(— ) dt^ = c p dt + mdp ; 



.-. dE 



42 



A 



dv 



But 






r~ - ^ J) j 



* + t2" 



dp. 



d*E 



dt dp dp dt 



dV ' c p 



. 



dpLA 



AT 



d Fm 
dtLA 



-m 



A\dp) f \dt, 



But 




= L[*?)-p 



d % v 




dp dt dt dp 



' ' A 



(dcA ldm\ -i __ (dv\ 



s dp 1 1 \dt I p_\ * dt) p 



(46) 



which is the relation between the thermal capacity m and the 
specific heat at constant pressure developed by the application 
of the first law to equation (6). 

Again, the same law may be applied to the equation (7). 



dQ = A(dE -\-pdv) = ndp + odv ; 
.-. dE = ?dp + (°-p)dv. . . 



Since 



d*E d*E 



(47) 



dp dv dv dp 

I idn\ 1 (do\ 

A\M p ~A\dpK 



1; 



fdo\ ldn\ 



A [_\dpj v \dvlp 



]- 



1. 



(48) 



FUNDAMENTAL EQUATIONS. 47 

Application of the Second Law — The second law of 

dQ 
thermodynamics is expressed by making -y, an exact differen- 
tial as indicated by equations (36) and (38). Applying this 
to equation (5) in the same way as was done with the first law, 

But 

[dQ ,, CdQ 



or 



dt dv ' ' dv dt j t j v dv dt 



d 1 c,.\ d t I \ 



dv\T)r dt\T} v 



r\dv r r 



—\ . (dfA - L , 



dt) v \dv) t T 



(49) 



the relation between / and c v developed by the application of 
the second law to equation (5). 

Applying to equation (6), we have 

d 1 c P \ d im 
~dp\T)rdt\f 



t<—\ 

1 fdc A \dt ). 



m 



A = 



T\dp ), T J 

/ dc j\ rdm \ m , N 

(w\~ far ~ t • • ■ • (50) 



4 8 



THERMO D YNAMICS OF THE STEAM-ENGINE. 



Again, applying to equation (7), 






d tn 

*'• ~dv\f) p ~dp\TJJ 



d 10 









^ 



<2fo 



I r rdt\ /dt\~ 

~TLWI~ *WWj "\ d P )-T\dv) t ' 



(50 



First and Second Laws Combined. — The result of ap- 
plying both the first and second laws of thermodynamics 
simultaneously to the fundamental equations is deduced by 
uniting the equations obtained by applying each separately. 

For the equation (5), in terms of c v and /, the comparison 
of equations (45) and (49) gives 



dp\ 1 / 

~dt) v = A T 



(52) 



For equation (6) the comparison of equations (46) and (50) 
gives 



idv\ 1 m 

\dt)p = ~ AT 



(53) 



For equation (7) the comparison of equations (48) and (51) 
gives 



A =~ 



r o( d ±\ 



T{_ \dp) v 



-©,] 



(54) 



or, substituting the values of n and from equations (17) and 

(is), 



<-=^),(S), 



(55) 



FUNDAMENTAL EQUATIONS. 49 

> 

Alternative Method. — Some writers on thermodynamics 
reserve the discussion of temperature until they are ready 
to define or assume an absolute scale independent of any 
substance, and depending only on the fundamental units of 
length and weight. Of the three general equations (5), (6), 
and (7) they use at first only the latter: 

dQ = ndp -f- odv. 

Now from the equation (23), representing the first law of 
thermodynamics, 

dQ = A(dB+pdv), 

it is evident that dQ is not an exact differential, since the 
equation cannot be integrated directly. The fact that in cer- 
tain cases/ may be expressed as a function of v, and the inte- 
gral for external work can be deduced, does not affect this 
general statement. Suppose that there is some integrating 

factor, which may be represented by — , so that 

may be integrated directly : we may then consider that we 
have a series of thermal lines represented by making 



— = const., — = const., ~^ T/ = const., etc. 



These lines with a series of adiabatic lines represented by 
q> = const., 0' = const., <p" = const., etc., 

allow us to draw a simple cycle of operations represented by 



5o 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Fig. 25, in which AB and CD are represented by the equa- 
tions 



— = C 9 and -~ f = C\ 



o a 




Fig. 25. 



while ^4Z> and BC are adiabatics. The ef- 

£-1 ficiency of a reversible engine receiving the 

heat Q during the operation AB, and re- 



jecting the heat Q during the operation CD^wiW be 



V — 



Q-Q AW 



Q 



Q ' 



dQ. 



But -~- is an exact differential, and depends on the state 

of the substance only, and consequently is the same at the end 
as at the beginning of the cycle, so that for the entire cycle 



5 



Now during the operations represented by the adiabatics 
AD and BC no heat is transmitted, and during the opera- 
tions represented by the lines AB and CD -=, is constant ; con- 
sequently the integration of the above equation for the cycle 
gives 

Q Q' 



Q-ffs-S' 



Q 



s 



that is, the efficiency of an engine working on such a 
cycle depends on 5 and S' , and on nothing else. 



FUNDAMENTAL EQUATIONS. 51 

Let us now define absolute temperature by making 

T=S, 
so that we have 

Q-Q T- T' 

Q ~ T ' 

and we will have a scale of temperature depending only on 
the efficiency of a reversible engine, and consequently inde- 
pendent of the properties of any substance. 

The discussion just given comes properly after the state- 
ment of the first and second laws of thermodynamics, and is 
followed by the application of those laws. 

Zeuner's Equations. — In his Mechanische Wdrmetheorie 
Zeuner employs the alternative method so far as to deducing 
equation (41). Then, instead of assuming that 5 is the abso- 
lute temperature, or giving such a definition of temperature, 
he assumes that the similarity of the thermodynamic equa- 
tions to certain gravitation equations indicates an essential 
similarity, and thereby avoids the second law of thermody- 
namics. Without discussing his method, there appears no 
reason why it might not be applied to deduce equations of 
the same form as those given on pages 44 to 48. He, how- 
ever, gives equations of a different form which may readily be 
deduced from our own, and which it may be convenient to 
write down here. Comparing equation (47) with 

, _ (dE\ idE\ 

dE =\Tp), d ^\Sv) t dv ' 



it is evident that 



n- fdE\ 
A ' ' \dplj 



A \ 



dE\ 

d~v)+*> 



$2 THERMODYNAMICS OF THE STEAM-ENGINE. 

These Zeuner writes: 

-(f) • 



Solving equation (54) for and for n y 

AT+nf£ 



<dt\ 



= 



dt\ 



n = 



-"■-'3), 



dv> p 

Substituting the value for in equation (7), we have 
1 r [dt\ ,. , idt\ 



dQ = 



But 






dt 



[-l^*+»(S*+^^]- 



= w*+ 



WW. 



afr; 






Again, substituting the value for n> 

dQ = -L \~o(^) dp + o{~) dv -A Tdpl ; 



(dt\ [_ \dpl v wv'j 

\dvjt 



FUNDAMENTAL EQUATIONS. 53 

Zeuner gives for his fundamental equations 

dQ = A{Xdp+ Ydv); 
A r„,. . /i 



dQ =m Xdt+ t« +t ) dv ~]' 

\dp) 



\dv) 

which may readily be deduced from equation (23) and the 
equations above. 

These equations are set down here because they are com- 
monly used by German writers in discussing thermodynamics. 



CHAPTER V. 
PERFECT GASES. 

The characteristic equation for a perfect gas is derived 
from a combination of the laws of Boyle and Gay-Lussac, 
which may be stated as follows : 

Boyle's Law. — When a given weight of a perfect gas 
is compressed (or expanded) at a constant temperature the 
product of the pressure and the volume is a constant. This 
law is nearly true at ordinary temperatures and pressures for 
such gases as oxygen, hydrogen, and nitrogen. Gases which 
are readily liquefied by pressure at ordinary temperatures, such 
as ammonia and carbonic acid, show a notable deviation from 
this law. The law may be expressed by the equation 

pv=p,v t , (56) 

in which p x and v x are the initial pressure and volume ; / is any 
pressure and v is the corresponding volume. 

Gay-Lussac's Law. — It was found by Guy-Lussac that 
any volume of gas at freezing-point increases about 0.003665 
of its volume for each degree rise of temperature. Gases 
which are easily liquified deviate from this law as well as 
from Boyle's law. In the deduction of this law temperatures 
were measured on or referred to the air-thermometer, and the 
law therefore asserts that the expansibility or the coefficient of 
dilatation of perfect gases is the same as that of air. It will 
be shown in this chapter that the scale of the air-thermometer 
differs but little from that of the absolute thermodynamic 

54 



PERFECT GASES. 55 

scale ; for practical purposes the difference may be neglected. 
Gay-Lussac's law may be expressed by the equation 

v = v (i +a)t, (57) 

in which v is the original volume at freezing-point, oc is the 
coefficient of dilatation or the increase of volume for one 
degree rise of temperature, and v is the volume corresponding 
to the temperature / measured from freezing-point. 

Coefficient of Dilatation. — Regnault* gives for the dila- 
tation from freezing- to boiling-point at Paris the results : 

Hydrogen 0.003667 

Atmospheric air 0.003665 

Nitrogen 0.003668 

Carbonic acid 0.003688 

In works on thermodynamics it has been commonly as- 
sumed that the coefficient of dilatation for air may be used for 
all gases, and at all temperatures and pressures, and that, con- 
sequently, on the centigrade scale a is 0.003665, or very 

nearly . Professor Holman f suggests that as the pressure 

273 
approaches zero the coefficient of dilatation of all gases 

approaches 

1 
a = , 

273.7 

which agrees with thermodynamic investigations relating to the 
absolute zero of temperature. On the Fahrenheit scale 

1 
a = . 

492.7 

Characteristic Equation. — From equation (57) we may 
calculate any special volume, such as v„ getting 

*\ = v »(\ + «)'• 

* Memoires de V Institut de France, tome xxi. 

f Lecture Notes on Heat, Mass. Inst. Technology. 



56 THERMODYNAMICS OF THE STEAM-ENGINE. 

Assuming that p x in equation (56) is the normal pressure of 
the atmosphere / , and substituting the value just found for v lf 
we have for the combination of the laws of Boyle and Gay- 
Lussac 



pv 



p v (i + a)t = A^o«(~ + t) . . (58) 



If now we assume that a perfect gas contracts part 

* 273-7 F 

of its volume at freezing-point for each degree decrease of 
temperature, then the absolute zero of temperature on the 
scale of the air-thermometer is 2J / $°.J C. below freezing-point; 
and centigrade temperatures can be transformed into absolute 
temperatures by adding 273°./. In like manner the zero of the 
air-thermometer on the Fahrenheit scale is 492°.7 below freez- 
ing-point, which on that scale is at 32 ; and absolute tem- 
peratures will be obtained by adding 460°./ to the Fahrenheit 
temperatures. 

The usual form of the characteristic equation for perfect 
gases may be derived from equation (58) by substituting T , 

the absolute temperature of freezing-point, for — , giving 



pv= P 4±T=RT, (59) 



where R is a constant representing the quantity 



T, 



For solution of examples it is more convenient to write 
equation (59) in the form 

/ T~" -y • a a o o o • (OO) 



PERFECT GASES. 57 

Absolute Temperatures. — For convenient reference it is 
desirable to write down the equations 

T = t -f- 273°.7 centigrade scale, 
T =. t -\- 46o°.y Fahrenheit scale, 

in which / is the ordinary temperature by the thermometer 
and T is the absolute temperature. 

Relation of French and English Units — Both French 
and English standard units of length and weight are arbitrary 
standards represented by the length of bars or the weight of 
certain pieces of metal. The French metre is the length 
from end to end of a bar of platinum at freezing-point; the 
English yard is the length from one line to another on a 
bronze bar at 62 F. ; owing to the different methods of de- 
fining the standards, comparisons are very difficult. The ratio 
of the lengths of the standards used in this book is that given 
by Professor Rogers,* namely, 

one metre = 39.3702 inches. 
< 
The last figure, 2, is probably uncertain to the extent of 
one unit. For engineering work the metre may be assumed 
to be equal to 39.37 inches. 

The ratio of the standards of weight f is 

one kilogram = 2.20462125 pounds. 

Acceleration due to Gravity. — The force with which a 
body is attracted towards the earth is proportional to the 
acceleration due to gravity, and varies with the latitude and 
with the altitude above sea-level, as given by the equation $ 

g = 980.6056 — 2.5028 cos 2 A. — 0.000003/2, . (61) 



* Proc. Am. Acad, of Arts and Sci., 1882-83; also additional observations, 
f Miller, Phil. Transactions, cxlvi, 1856. 
X Everett's Units and Phys. Const. 



58 THERMODYNAMICS OF THE STEAM-ENGINE. 

in which g is the acceleration in centimetres per second, A is 
the latitude, and h is the altitude above the sea in centimetres. 

Standard Latitude. — It is customary to reduce all physi- 
cal observations which are affected by the attraction of gravity 
to the latitude of 45 and to sea-level. This reduction affects 
the fifth significant figure, and consequently is properly 
recognized in recording physical constants which may be 
used by engineers, but it is seldom, if ever, necessary for an 
engineer to reduce such physical constants to the place where 
he may be located. 

Specific Pressure. — The normal pressure of the atmos- 
phere is assumed to be equivalent to that of a column of 
mercury 760 mm. high at freezing-point. Now Regnault * 
gives for the weight of a litre, or one cubic decimetre, of 
mercury 13.5959 kilograms; consequently the specific pressure 
of the atmosphere under normal conditions is 

p o = 10333 kilograms per square metre. 

Using the conversion units given above for reducing this 
specific pressure to the English system of units gives 

p = 2146.32 pounds per square foot, 

which corresponds to. 

14.6967 pounds per square inch, 
or to 

29.921 inches of mercury. 

It is customary and sufficient to use for the pressure of the 
atmosphere 14.7 pounds to the square inch. 

* Mhnoirei de V Institut de France, vol. xxi. 



PERFECT GASES. 59 

Densities of Gases. — The weights in kilograms of one 
cubic metre of several gases, as determined by Regnault, are 
given in the following table: 

Atmospheric air 1.293 187 

Nitrogen 1. 256167 

Oxygen 1 .429802 

Hydrogen 0.089578 

Carbonic acid 1.9774 14 

Now the acceleration due to gravity at Paris, as calculated 
by equation (61), is 980.9218 centimetres, while the acceler- 
ation at 45 latitude and at sea-level is 980.6056 centimetres; 
consequently the pressure due to 760 mm. of mercury at 45 
latitude is equivalent to that of only 

080.6056 
760 X fg^TS = 759 - 755mm - 

at Paris, and the weights of one cubic metre of the several gases 
at the normal pressure of the atmosphere will be smaller in a 
like proportion. To get the specific volumes we may multiply 
the weights by the ratio 

759-755 
760 ' 

and then take the reciprocal of the results, giving results set 
down in the following paragraph. 

Specific Volumes. — The following table gives the specific 
volumes of several gases in cubic metres per kilogram : 

VOLUMES IN CUBIC METRES OF ONE KILOGRAM 
AT 45° OF LATITUDE. 

Atmospheric air °-77353 2 7 

Nitrogen 0.7963291 

Oxygen 0.69962 3 1 

Hydrogen 11.16705 

Carbonic acid 0.5058741 



60 THERMODYNAMICS OF THE STEAM-ENGINE. 

The corresponding quantities for English units are given 
in the next table : 

VOLUMES IN CUBIC FEET OF ONE POUND AT 45 ° OF LATITUDE. 

Atmospheric air 12.3909 

Nitrogen 12.7561 

Oxygen 1 1 .2070 

Hydrogen „ 178.881 

Carbonic acid 8. 10324 

Value of R. — The constant R which appears in the usual 
form of the characteristic equation for a gas represents the 
expression 

T ' 

X 

The values for R corresponding to the French and the 
English system of units may be calculated as follows: 

French units, R = io 333 X Q-77353 = 2 ^ / 62) 

273-7 

English units, R = 2Il6 -3 X 12.391 = ( 

s 492.7 DD v 0J 

Value of R for other gases may be calculated in a like 
manner. 

Solution of Problems. — Many problems involving the 
properties of air or other gases may be solved by the charac- 
teristic equation 

pv = RT y 

or by the equivalent equation 

pv _ p v 
T~ T,' 

which for general use is the more convenient. 



PERFECT GASES. 6 1 

In the first of these two equations the specific pressure and 
volume to be used for English measures are pounds per square 
foot, and the volume in cubic feet of one pound. 

For example, let it be required to find the volume of 3 
pounds of air at 60 pounds gauge-pressure and at ioo° F. 
Assuming a barometric pressure of 14.7 pounds per square 
inch, 

53.22(460.7+ 100) 

v == ; , , \ = 2 '774 cubic feet 

(14.7+60)144 "* 

is the volume of 1 pound of air under the given conditions, 
and 3 pounds will have a volume of 

3 X 2.774 = 8.322 cubic feet. 

The second equation has the advantage that any units may- 
be used, and that it need not be restricted to one unit of 
weight. 

For example, let the volume of a given weight of gas, at 
100° C. and at atmospheric pressure, be 2 cubic yards; re- 
quired the volume at 200 C. and at 10 atmospheres. Here 

\ov 1X2 



4737 ' 373-7 ' 

473-7 X 2 

v = — = 0.2535 cubic yards. 

IOX3737 

Application of the Laws of Thermodynamics. — Equa- 
tion (55), page 48, 

<dv\ (dp^ 






which was deduced by applying both laws of thermodynamics 
to equation (7), may be most conveniently used in this place. 



62 THERMODYNAMICS OF THE STEAM-ENGINE. 

Differentiating the characteristic equation 

pv = R T, 
we obtain 



dv\ R 



dt' v v 



which substituted in equation (55) gives 

c p — c v — AR (64) 

Specific Heat at Constant Pressure — The specific heat 
for true gases is very nearly constant, and may be considered 
to be so for thermodynamic equations. Regnault gives for 
the mean values for specific heat at constant pressure the fol- 
lowing results : 

Atmospheric air 0.2375 

Nitrogen 0.2438 

Oxygen 0.2175 

Hydrogen 3-409 

Specific Heat at Constant Volume. — It is evident from 
inspection of equation (64) that the specific heat at constant 
volume is a constant, and that equation also gives one of the 
best ways of calculating this quantity. Most commonly the 
ratio of the two specific heats is determined, as follows: 



K= cr~z^ (65) 



K = 



i _ 10333 x 0.77353 

426.9 X 273.7 X 0.2375 
k = 1.4046 = 1.405, nearly. 



PERFECT GASES. 63 

Thermal Capacities. — Substituting the values of the par- 
tial differential coefficients as deduced from equation (59), in 
equations (n), (15), 17), and (18), we have for the values of 
the thermal capacities for gases 



1 -r^ c p~ c J = ^ c p — *») ; • • • ( 66 ) 



v T 

m — — -g(c p - c v ) = — — (c p - c v ) ; . . (67) 



v T 

n = R C * = -p C "> ( 68 ) 



P T 

~ R Cp = v C *' ' ' " ^ 



General Equations. — To deduce the general equations 
for gases from equations (5), (6), and (7) it is only necessary 
to replace the letters /, m, n y and by their values just ob- 
tained, giving 

T 

dQ = c v dt -f- fa — Cv)~ dv ; . . . . (70) 



T 

dQ = c P dt + (<:„ — ^)-^> ; . . . . (71) 



T T 

dQ = c—dp + c p —dv. . . o c . (72) 



64 THERMODYNAMICS OF THE STEAM-ENGINE. 

Isothermal Line — The equation to the isothermal line 
for a gas is obtained by making T a constant in the character- 
istic equation, so that 

pv — RT, = p 1 v l , 
or 

pv=Pf>* (73) 

This equation will be recognized as the expression of 
Boyle's law. It is, of course, the equation to an equilateral 
hyperbola. 

To obtain the external work during an isothermal expan- 
sion we may substitute for/ in the expression 



W 



= j pdv 



from the equation to the isothermal line just stated, using 
for limits the final and initial volumes, z\ and v x : 

itt f v,> dv . v. , x 

W = J> l v, I — = /,*, log, -*. . . . (74) 

If the problem in any case calls for the external work of 
one unit of weight of a gas, then %\ and v t must be the initial 
and final specific volumes; but in many cases the initial and 
final volumes are given without any reference to a weight, 
and it is then sufficient to find the external work for the given 
expansion from the initial to the final volume without con- 
sidering whether or not they are specific volumes. 

The pressures must always be specific pressures ; in the 
English system the pressures must be expressed in pounds on 
the square foot before using them in the equation for external 
work or, for that matter, in any thermodynamic equation. 

For example, the specific volume of air at freezing-point and 
at 14.7 pounds pressure per square inch is about 12.4 cubic 
feet ; at the same temperature and at 29.4 pounds per square 
inch the specific volume is 6.2 cubic feet. The external work 



PERFECT GASES. 6$ 

during an isothermal expansion of one pound of air from 6.2 
to 12.4 cubic feet is 

W = p x v, / — = p,v x log, - 

12. A 

= 29.4 X 144 X 6.2 log, — — = 18 190 foot-pounds. 

For example^ the external work of isothermal expansion 
from 3 cubic feet and 60 pounds pressure by the gauge to a 
volume of 7 cubic feet is 

7 
W— (60+ 14.7)144 X 3 l°g* - = 2 7340 foot-pounds. 

In both problems the pressure per square inch is multi- 
plied by 144 to reduce it to the square foot. In the first 
problem the pressures are absolute, that is, they are measured 
from zero pressure ; in the second problem the pressure by 
the gauge is 60 pounds above the pressure of the atmosphere, 
which is here assumed to be 14.7 pounds per square inch, and 
is added to give the absolute pressure. In careful ex- 
perimental work the pressure of the atmosphere is measured 
by a barometer and is added to the gauge-pressure. 

Isoenergic Line. — The isothermal line for a perfect gas 
is also the isoenergic line, a fact that may be proved as 
follows: The heat applied during an isothermal expansion 
may be calculated by making T a constant in equation (70) 
and then integrating; thus: 

Q = fy, - c,)T, P^ = (c, - c.) T, log, ft , 

or, substituting for c p — c v from equation (64), 

Q= ART, log, v -i = Ap lVl log,?i. . . (75) 

V, V, 




06 THERMODYNAMICS OF THE STEAM-ENGINE. 

A comparison of equation (75) with equation (74) shows 
that the heat applied during an isothermal expansion is equiv- 
alent to the external work, or we may say that all the heat 
applied is changed into external work, so that the intrinsic 
energy is not changed. 

An interesting corollary of the discussion just given is that 
an infinite isothermal expansion gives an infinite amount of 
work. Thus the area contained between 
the axis OV (Fig. 26), the ordinate ab, and 
the isothermal line aa extended without 
limit is 

rir 1 °° 

W = p v log,— = 00. 
Fig. 26. v 

This may also be seen from the consideration that if heat 
be continually applied, and all changed into work, there will 
be a limitless supply of work. 

Adiabatic Lines. — During an adiabatic change — for exam- 
ple, the expansion of a gas in a non-conducting cylinder — heat 
is not communicated to, nor abstracted from, the gas; conse- 
quently dQ in equations (70), (71), and (72) becomes zero. 

From equation (72) 

o = dQ=^vdp+ C £pdv; 

c p dv _ dp # 
. . — — , 

c v v p 



.-. io & e)-=io g M 



The ratio — of the specific heats may be represented by k, 
and the above equation may be written 

...... (76) 



£M< 



P 

v K p — v*p x = const (77) 



PERFECT GASES. 6? 

This is the adiabatic equation for a perfect gas which is 
most frequently used. If adiabatic equations involving other 
variables, such as v x and T lt are desired, they may be derived 
from equation (76) by aid of the characteristic equation, which 
for this purpose may be written 

/ pv p x v x 

so that 

A __ *>T X 
P v x T % 

and 

tv\ K ~ x T 

-) = j; ...... . ( 7 8) 

.-. 7V"- 1 = T x v?~ x (79) 

Or equations (78) and (79) may be deduced directly from 
equation (70) as equations (76) and (77) were from equa- 
tion (72). 

In like manner we may deduce from equation (71) 



I — < T — K 



tp « = t;a < , (80) 

or we may derive it from equation (76). 
To find the external work the equation 

W= f pdv 

maybe used after substituting for/ from equation (77): 

W 

PJ>x 



= ["pdv = vfPx F*L = - ASL(_i L_) . 



Wz 

K — I 



68 THERMODYNAMICS OF THE STEAM-ENGINE. 



\ 



a 



In Fig. 27 the area between the axis OV, 
the ordinate ba, and the adiabatic line aa ex- 
tended without limit becomes 

W - piV * 

1 ~ ~ > 

Fig. 27. K — l 

and not infinity, as is the case with the isothermal line. 

Here, as with the calculation of external work during iso- 
thermal expansion, specific volumes should be used when the 
problem involves a unit of weight ; but work may be calcul- 
ated for any given initial and final volumes without consider- 
ing whether they are specific volumes or not. The pressures 
are always pounds on the square foot for the English 
system. 

For example, the external work of adiabatic expansion 
from 3 cubic feet and 60 pounds pressure by the gauge to the 
volume of 7 cubic feet is 

W= 74-7 X 144 X3j , _ (i) ,405 "'| = 23I4 o foot-pounds, 

which is considerably less than the work for the correspond- 
ing isothermal expansion. 

Intrinsic Energy. — Since external work during an adia- 
batic expansion is done at the expense of the intrinsic 
energy, the work obtainable by an infinite expansion can- 
not be greater than the intrinsic energy. If it be admit- 
ted that such an expansion changes all of the intrinsic energy 
into external work we shall have 

E x = W x = J&-, . . . . (82) 

K — I V ' 

which gives a ready way of calculating intrinsic energy. In 
practice we always deal with differences of intrinsic energy, 
so that even if there be a residual intrinsic energy after an 



PERFECT GASES. 69 

infinite adiabatic expansion the error of our method will be 
eliminated from an equation having the form 

E,-E,= *&-- -&-. • • • (83) 

1 a K — I K — I 

Exponential Equation. — The expansions and compres- 
sions of air and other gases in practice are seldom exactly 
isothermal or adiabatic ; more commonly an actual operation 
is intermediate between the two. It is convenient and 
usually sufficient to represent such expansions by an exponen- 
tial equation, 

fif=P 1 v 1 u , (84) 

in which n has a value between unity and 1.405. The form 
of integration for external work is the same as for that of 
adiabatic expansion, and the amount of external work is inter- 
mediate between that for adiabatic and that for isothermal 
expansion. 

Change of temperature during such an expansion may be 
calculated by the equations 

Tf-*=T x v?-\ (85) 

which may be deduced from equation (82) by aid of the 
characteristic equation 

pv = RT 

as equation (79) is deduced from equation (76). 

If it is desired to find the exponent of an equation repre- 
senting a curve passing through two points, as a x and a % (Fig. 
28), we may proceed by taking logarithms of 
both sides of the equation 



giving 

n log v x + log/, = n log v 9 + log p„ Fig. 2s. 



p 


d, 


i 3 







V 



70 THERMODYNAMICS OF THE STEAM-ENGINE. 

so that 

log A - log A , Q „v 

* = ; i • ( 8 7) 

log v 3 — log v l 

For example, the exponent of an equation to a curve pass- 
ing through the points 

A = 74- 7> ^ = 3> and /, = 30, v, = 7 



is 



log 74-7 - log 30 
n — — = 1. 104. 

log 7 — log 3 



Entropy — For any reversible process 



^ = f; 



consequently from equations (70), (71), and (72) we have 



dt dv 



dt dp 

dcf) = ^y + (<:„ = c p ) ~ , 



^0 = C — -|_ C ; 



and, integrating between limits, 



T v^ 

<P* - 0, = *. log, ^ + fo - O log, ~, . (88) 



PERFECT GASES. 7 1 

0. - 0i = c t log, j. + (*> - ^) log, — , . (89) 



A v i 

<P* - <?>> = c v log, — + c p log, -, . . . (90) 

Px 1 



which give ready means of calculating changes of entropy. 

For example, the change of entropy in passing from the 
pressure of 74.7 pounds absolute per square inch and the 
volume of 3 cubic feet to the pressure of 30 pounds absolute 
and the volume of 7 cubic feet is 



°- 2 375 ,3° ,7 

02 ~ 01 = T^5~ ge 74.7 + °' 2375 lo ^3 = -°454. 



Since the pressures form the numerator and denominator 
of a fraction, there is no necessity to reduce them to the 
square foot. In this problem the pressures and volumes are 
taken at random ; they correspond to a temperature of 146 
F. at the initial condition. 

Comparison of the Air-thermometer with the Absolute 
Scale — In connection with the isodynamic line it was shown 
that the intrinsic energy is a function of the temperature only. 
This conclusion is deduced from the characteristic equation 
on the assumption that the scale of the air-thermometer coin- 
cides with the thermodynamic scale, and it affords a delicate 
method of testing the truth of the characteristic equation, and 
of comparing the two scales. 

The most complete experiments for this purpose were 
made by Joule and Sir William Thomson, who forced air 
slowly through a porous plug in a tube in such a manner that 
no heat was transmitted to or from the air during the process. 
Also the velocity both before and beyond the plug was so 
small that the work due to the change of velocity could be 
disregarded. All the work that would be developed in free 



72 



THERM OD YNAMICS OF THE STEAM-ENGINE. 



expansion from the higher to the lower pressure was used in 
overcoming the resistance of friction in the plug, and so con- 
verted into heat, and as none of this heat escaped it was re- 
tained by the air itself, the plug remaining at a constant tem- 
perature. It therefore appears that the intrinsic energy re- 
mained the same, and that a change of temperature indicated 
a deviation from the assumptions of the theory of perfect 
gases. 

In the discussion of results given by Joule and Thomson * 
in 1854 they give for the absolute temperature of freezing- 
point 273°.7 C. As the result of later f experiments they 
state that the cooling for a difference of pressure of 100 inches 
of mercury is represented on the centigrade scale by 



o .92^—^- 



The following table shows the agreement between this 
statement and the results of experiment : 

FLOW OF AIR THROUGH POROUS PLUG: 



Temperature. 


Cooling Effect : 


By Experiment. 


By Calculation. 


o° 

7.1 

39-5 
92.8 


O.92 
O.88 
0.75 
O.51 


O.92 
O.87 
O.70 
O.51 



From the work of these experiments Rowland \ deduced 
the following comparison of the air-thermometer at constant 
volume with the absolute thermodynamic scale of temperature : 



* Philosophical Transactions •, vol. 144, p. 349. 

f Ibid., vol. 152, p. 579. 

% Proceedings of the American Academy, vol. xv (N. S. viii), p. 75, 1879. 



PERFECT GASES. 

REDUCTION OF THE AIR-THERMOMETER TO THE 
ABSOLUTE SCALE. 

{Centigrade.) 



73 



Temperature above Freezing. 


Correction to Air- 
thermometer. 


Air-thermometer. 


Absolute Scale. 


o° 

10 

20 

30 
40 
50 
60 
70 
80 
90 

100 
200 
300 
400 
500 




9.9972 
19.9952 
29.9939 

39-9933 
49.9932 

59-9937 
69.9946 

79-9956 
89.9978 
100.000 
200.037 
300.092 

400.157 

500.228 




— O.OO28 

— O.OO48 

— 0.0061 

— O.O067 

— 0.0068 

— O.O063 

— O.O054 

— O.O044 

— 0.0022 
O. 

+ O.037 
+ O.O92 
+ O.157 
-f- O.228 



C AB 



.C 



•m?- 



Fig. 29. 



Velocity of Sound. — Sound is transmitted through the 
air in spherical waves, but at a distance from the source of 
sound the waves are sensibly plane waves, 
and the progress of the wave is the same 
as that of a plane wave in a straight tube ■ 
of uniform section. Let Fig. 29 repre- 
sent a tube one square metre in section in which a wave 
moves with a linear velocity u metres per second ; that is, a 
point at a given phase of the wave — for example, C at the 
greatest condensation — moves at that velocity. 

Since the wave moves with the velocity « o , the volume of 
air disturbed in a unit of time is u cubic metres. If the 
specific volume in the undisturbed state is v , then the weight 
of air disturbed in a second is 



w = me = — , 



m being the mass of air which has the weight w. 



74 THERMO D YNAMICS OF THE STEAM-ENGINE. 

Imagine two planes A and B at a small distance apart, which 
also move with the velocity u , so that they remain at the same 
phase of the wave. Let the absolute velocities of the air at 
these planes be u x and u^ ; then the velocities of the air 
through the planes — that is, the velocities relatively to the 
planes — is, for A, u —u Jt and for B, ti — u^. With v t and 
v 9 for the specific volumes at these planes the weights that 
pass through the planes A and B per second are 

l ^^l and *CZ*. 
v l v % 

Since the phase of that portion of the wave between A and 
B is constant, the weight of the air between them is also con- 
stant, and as much air enters per second as leaves during that 
time. Again : as, on the whole, the air is not transmitted, but 
only compressed and rarefied, the whole air disturbed per 
second must pass through the space between the planes. 
Therefore 

u — u. u — #„ u n 

-? » — ■ =z -£ = mg\ 

.-. Ul = u — mv x g\ 
u, = u — mv,g; 
«i — «i = mgtyt — v,). 

Now as the mass m enters the space between the planes 
with the absolute velocity u x , and an equal mass leaves with 
the velocity « a , consequently there is a change of momentum 

m{ii x — u^) ; 

and since this cannot come from the mutual action of the par- 
ticles, it must come from the difference of pressures at A and 
B\ thus: 

A - A = **(*i - « a ) = ^VO, — *>,)• 



PERFECT GASES. 75 

As the planes A and B approach each other p x and/ a , v x 
and v 3 approach in value, and at the limit 

dp = — m*gdv, 

dp__ , l_*J| 

^_ «^_ ^^ ff 

the last reduction being obtained by substituting for the value 
of m from the preceding work. Solving for u , 

dp 

The vibrations are so rapid that the changes of state may 
be assumed to be adiabatic ; consequently equation (72) gives 

o = dQ = C -£vdp + C pdv: 



-^- = - Cp - - = — /c-. 

dv c v v v 



The planes A and B may be taken at any phase of the 
wave ; for example, at the phase where the pressure and vol- 
ume are normal, in which case 

dp A 

-r- = — AT—. 

dv v Q 

Substituting in the equation for u i% we have 

U* = gKZ\p,. 

The equation is commonly given in terms of the density, 
y } as follows : 

U 9 =.A/KgL, ...... (91) 



y6 THERMODYNAMICS OF THE STEAM-ENGINE. 

The velocity of sound from direct experiment was found 
by Moll and Van Beek to be 332.26 metres per second; by 
Regnault to be 330.70 metres per second. Kayser found 
from Kundt's dust figures the wave length corresponding to a 
certain tone, and therefrom deduced the velocity of sound, and 
gives for the velocity 332.50 metres per second. The true 
value must be nearly 332 metres per second. Solving equa- 
tion (91) for /c, and inserting the known values of p v Q and g 
for Paris, 



K 



«•" 332* 



gv % p % 9.8092 X 0.77328 x 10333 ' 

K = I.4063. 

This result as calculated compares very well with that 
already calculated by aid of equation (65), but it depends 
entirely on the value assigned to the velocity of sound 
whether the comparison is satisfactory or not. Now that the 
determination of the mechanical equivalent of heat by Row- 
land has reduced the probable error of that physical constant 
to a small amount the calculation by equation (65; is to be 
preferred, especially as it agrees very well with the results of 
direct experiments by Rontgem 

Rontgen's Experiments, — Direct experiments to deter- 
mine k may be made as follows • Suppose that a vessel is 
filled with air at a pressure p t , while the pressure of the 
atmosphere is p a , Let a communication be opened with the 
atmosphere sufficient to suddenly equalize the pressure; then 
let it be closed, and let the pressure / 9 be observed after the 
air has again attained the temperature of the atmosphere.. 
If the first operation is sufficiently rapid it may be assumed 
to be adiabatic, and we may use equation (77), from which 



K ^QgA -logA / v 

log Va — log ^ * 



PERFECT GASES. J J 

The second operation is at constant volume ; consequently 
the specific volume is the same at the final state as after the 
adiabatic expansion of the first operation. But the initial and 
final temperatures are the same ; consequently 

v,p x = v a p 7 ; 

.-. log v a — log v x — log/, - log A, 

which substituted in equation (92) gives 

log* -jog A 

log A - log A w/ 

The same experiment may be made by rarefying the air in 
the vessel, in which case the sign of the second term changes. 

Rontgen * employed this method, using a vessel contain- 
ing 70 litres, and measuring the pressure with a gauge made 
on the same principle as the aneroid barometer. Instead of 
assuming the pressure A at the instant of closing the stop-cock 
to be that of the atmosphere, he measured it with the same 
instrument. A mean of ten experiments on air gave 

K = L4053. 

The value which will be used in this book for the ratio of 
the specific heats of air is 

K = ^ = I.4O5. 
EXAMPLES. 

1 . Find the weight of 4 cubic metres of hydrogen at 30 C 
and under the pressure of 800 mm. of mercury. Ans. 0.3398 

kg. 

2. Find the volume of 3 pounds of nitrogen at a pressure 
of 45 pounds to the square inch by the gauge and at 8o° F. 
Ans. 10.34 cu. ft. 

* Poggendorff's Annalen, vol. cxlviii, p. 580. 



78 THERMOD YNAMICS OF THE STEAM-ENGINE. 

3. Find the temperature at which one kilogram of air will 
occupy one cubic metre when at a pressure of 20,000 kilograms 
per square metre. Ans. 411°. 2 C. 

4. Find the pressure at which 2 pounds of carbonic acid 
at freezing-point of water will occupy 3 cubic feet.^ Ans. 79.4 
pounds per sq. in. 

5. A gas-receiver having the volume of 3 cubic feet con- 
tains half a pound of oxygen at 70 F. What is the press- 
ure? Ans. 29.6 pounds per sq. in. 

6. A spherical balloon 20 feet in diameter is to be inflated 
with hydrogen at 6o c F. when the barometer stands at 30.2 
inches, so that gas may not be lost on account of expansion 
when it has risen till the barometer stands at 19.6 inches and 
the temperature falls to 40 F. How many pounds and how 
many cubic feet are to be run in? Ans. 15. 1 pounds, 2828 
cu. ft. 

7. A gas-receiver holds 14 ounces of nitrogen at 20° C. 
and under a pressure of 29.6 inches of mercury. How many 
will it hold at 32 F. and at the normal pressure of 760 mm.? 
Ans. 15. 18 oz. 

8. Two cubic feet of air expand at 50 F. from a pressure 
of 80 pounds to a pressure of 60 pounds by the gauge. What 
is the external work? Ans. 6464 ft. lbs. 

9. What would have been the external work had the air 
expanded adiabatically ? Ans. 4450 ft. -lbs. 

10. Find the external work of 2 pounds of air which ex- 
pand adiabatically until it doubles its volume, the initial 
pressure being 100 pounds absolute and the initial tempera- 
ture ioo° F. Ans. 36,100 ft. -lbs. 

11. Find the external work of one kilogram of hydrogen 
which, starting with the pressure of 4 atmospheres and with 
the temperature of 500 C, expands adiabatically till the 
temperature becomes 30 C. Ans. 489,280 m.-kg. 

12. Find the exponent for an exponential curve passing 
through the points/ = 30, v = 1.9, and p x = 15, v x = 9.6. 
Ans. 0.4279. 



PERFECT GASES. 79 

13. Find the exponent for a curve to pass through the 
points/ = 40, i/=2, and p x = 12, x\ = 6. Ans. 1.0959. 

14. In examples 2 and 3 let/ be the pressure in pounds 
on the square inch and v the volume in cubic feet. Find the 
external work of expansion in each case. Ans. 21,900 and 
12,010 ft. -lbs. 

15. Find the intrinsic energy of one pound of nitrogen 
under the standard pressure of one atmosphere and at freezing- 
point of water. Ans. 66,655 ft. -lbs. 

16. A pound of air has the volume 6 cubic feet under the 
pressure of 30 pounds absolute to the square inch. Find the 
intrinsic energy. Ans. 64,000 ft. -lbs. 

17. In example 16 find the increase of entropy above 
that at atmospheric pressure and at freezing-point. Ans. 
— 0.0516. 

18. A kilogram of oxygen at the pressure of 6 atmos- 
pheres and at ioo° C. expands isothermally till it doubles its 
volume. Find the change of entropy. Ans. 40.0434. 

19. Suppose a hot-air engine, in which the maximum 
pressure is 100 pounds absolute, and the maximum temper- 
ature is 6oo° F., to work on a Carnot's cycle. Let the 
initial volume be 2 cubic feet, let the volume after isothermal 
expansion be 5 cubic feet, and the volume after adiabatic 
expansion be 8 cubic feet. Find the external work of one 
cycle ; also the horse-power if the engine is double-acting 
and makes 30 revolutions per minute. Ans. 8966 ft. -lbs., 
16.3 H.P. 



CHAPTER VI. 
SATURATED VAPORS. 

OUR knowledge of the properties of saturated vapors is 
derived mainly from the experiments of Regnault.* In most 
works on steam and other vapors the results of his experi- 
ments are given in the form of empirical equations designed 
to be used for calculating tables of the properties of vapors. 
Many such tables have been calculated, and unfortunately 
they show such diversity of values as to lead to serious incon- 
venience when applied in practice. It therefore appears 
advisable to give the original data on which the empiricaL 
equations are based, so as to exhibit the limits of their 
application and the degree of accuracy to be attributed to 
them. The constants for the equations for calculating the 
pressures of saturated steam at different temperatures have 
been recomputed with care and accuracy, because there are 
minor discrepancies in the equations given by Regnault. 
Rowland's determinations of the mechanical equivalent of heat 
and of the specific heat of water have been adopted, and 
all of Regnault's results depending thereon have been revised 
and brought into concordance with them. 

Pressure of Saturated Vapor. — Regnault's experiments 
on the temperature of saturated vapor consisted essentially in 
taking the temperature of the boiling-point of the vapor under 
varying pressures of the atmosphere, the apparatus being so 
arranged that the pressure could be varied from a small frac- 
tion of an atmosphere to more than twenty atmospheres. 

* MJmoires de V Institut de France, etc., tome xxvi. 

80 



SA T URA TED VAPORS. 



81 



The temperature was taken with mercurial thermometers, 
and the pressures were measured by a mercury column, and, 
after the necessary corrections were applied and temperatures 
were reduced to the air-thermometer, Regnault selected the 
results he deemed most trustworthy, and plotted a series of 
points, and then drew a smooth curve to represent the whole 
series of experiments. 

He then selected points on the experimental curve at regu- 
lar intervals, and with these points as data he calculated the 
constants of empirical formulae for use in calculating the 
tabular values. The formula selected was of the form 

log p — a + ba n -f c/3 n , .... (94) 

in which p is the pressure, and n is the temperature minus 
the constant temperature t of the lowest limit of the range of 
temperature to which the formula applies; i. e., 

n = t — /,. 

Let the points through which the curve represented by 
the equation is to pass be (p , /„), (p l9 A), (p %i Q, (/,, /,), 
and (/> 4 , / 4 ), so chosen that 

.-. t % -* % = 2(/ I -/ i ), (/ 3 -0 = 3(A-'.)» ('«-'.) =4(',-0. 

Substituting the five known values of p and t in equation (94), 



logp 9 = a + b +c; 

log p x — a -\- ba' 1 '* -^- c/3*i-'°; 
\ogp 2 = a + ba 2 ^-^ -f- cp*'*-** ; 
logp 3 = a -f ba^-^ -f- c/3&i-*°) ; 

log/ 4 = «: + ^«4(/i-/ ) _|_ c/34d-'»K 



• (95) 



82 



THERMO D YNAMICS OF THE STEAM-ENGINE. 



Now subtract each equation, member for member, from 
the one below it, and for convenience let 

l°gA — l°gA — y* » etc -> « /l ~ / ' = m, fi*> -*» = n. 



y. 


= (?^ 


— i)£ -[- (» — i)c; " 


yx 


= (m* 


— m)b + (n* — ri)c ; 


y* 


= (m 3 


— m*)b -f- (/z 3 — /z 2 )<: ; 


y* 


= (^* 


— m*)b -j- (V — « 3 )^. J 



• (96) 



Solving the several equations for c and equating the 
values, 

y — (m — i)b y v — {m 7 — m)b 
n — 1 ri* — n 



y — (m % — m*)b y 2 — (m — m)b 



3 2 
n — n 



n* — n 3 



• • (97) 



Again, solving for b and equating the values and reduc- 



ing, 



ny Q —y x ny x —y^ 



ny. z - y 3 



n — m {n — ni)m (n — m)if^' 



m ny — m y v = mny y — my^ = ny 2 — j/ s , 



mny — my x = ny y — jv> 



mny x — my % = ny^ — y^ 



(m + n)y, — y, (m + n)y 9 — y 3 

mn = = ; 



(98) 



m 



4- n = — a = M ; 

j\ — y»y* 



y * —y ^y* AT 

mn = —i = N. 



yx —y y, 



(99) 
(100) 



SA T URA TED VA P ORS. 8 3 

Equations (99) and (100) enable us to calculate numerically 
the values of M and N from the five given values of log p. 
Then solving for m and ;z, 



M /M 2 V 
m = — 

2 \ < 



(?"-)': 



„ = - + {— - N ). 



Solving one of the equations (97) for b> 



= —, r ~, — a \ — 7 \? \« U 01 ) 

nym — 1) — (m — m) {in — \){n — m) J 



Again, solving the first equation of (96) for c, 

_ ny — y x 

n — m y. — my a , x 

c = = ■ , Jl ., - /0 , . . (102) 

it — 1 (n — i)(n — m) 

From the first equation of (95) 

a = logp -b-c (103) 

Finally, a = m** ~ /o ; (104) 

1 
ft = it**-'** ; (105) 

For temperatures below freezing-point Regnault used the 
equation 

p — a-\- &a n , ( IQ 6) 

which is an equation to a curve passing through three points 
at equidistant temperatures, and of which the solution is very 
simple. 



8 4 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Regnault's Data and Equations for Steam. — For equa- 
tion (106) the data are: 

t = — 32 C. p = 0.32 mm. of mercury. 

t x — - i6°C. p i = 1.29 " 

t 9 = o°C. A = 4.60 " " " 

From which Regnault calculated the following equation by 
aid of seven-place logarithms : 

A. For steam from — 32 to o° C, 

p = a + ba n \ 

a = — O.08038; 
log b = 9. 6024724 — 10; 
log a= 0.033398; 

n = 32 — t. 

Regnault uses three equations of the form given by equa- 
tion (97), for which the following are the data: 



B. 



C. 



D. 



<.= o° 


c. 


A 


= 


4.60 mm. of mer 


t s = 25 


c. 


A 


= 


23-55 ' 




;, = 50 


c. 


A 


= 


91.98 ' 




'3= 75° 


c. 


A 


— 


288.50 ' 




/ 4 =- IOO° 


c. 


A 


= 


760 ' 




t = IOO° 


c. 


A 


= 


760 ' 




', = 1 30° 


c. 


A 


= 


2030 ' 




t % = 160 


c. 


A 


= 


4651.6 




*, = 190 


c. 


A 


— 


9426 ' 




t t = 220° 


c. 


A 


= 


17390 




t =z - 20' 


D c. 


A 


= 


0.91 ' 




*, = + 40 


°c. 


A 


= 


54-9 1 ' 




t % = IOO° 


c. 


A 


= 


760 4 




/, = 160 


c. 


/a 


= 


4651.6 * 




t K = 220° 


c. 


A 


= 


17390 





SATURATED VAPORS. 85 

And from these data he calculated, by aid of seven-place 
logarithms, the following equations, which are correct at Paris : 

B. For steam from o° to ioo° C, 

log p = a — ba n -j- cft n ; 

a = 4.7384380; 
log b — 0.6116485 ; 
log c = 8.1340339 — 10 ; 
log a = 9.9967249 — 10; 
log fS = 0.006865036 ; 

n = t. 

C. For steam from ioo° to 220 C, 

log p = a — bct n -f cfi n ; 

a = 5.4583895 ; 

log b = 0.4121470 ; 
log c = 7.7448901 — 10; 
logar= 9.997412127 — 10 ; 
log/5 = 0.007590697 ; 
» = t — 100. 

D. For steam from — 20 to 220 C, 

log/ = a — ba n — c/3 n ; 

a = 6.2640348 ; 
log b = 0.1397743 ; 
log c = 0.6924351 ; 
log a= 9.994049292 — 10; 
log/3 = 9.998343862 — 10 ; 

« = /-}- 20. 

Regnault gives complete tables of the properties of satu- 
rated steam, calculated by the equations just set down, using 
equations A and B for their respective ranges of temperature, 
but he uses equation D (which applies to the entire range 
from — 20°C. to -(- 120 C.) instead of equation C. He prob- 



86 THERMODYNAMICS OF THE STEAM-ENGINE, 

ably did this because the constants calculated for equation C 
with seven-place logarithms are unsatisfactory. None of the 
constants calculated by seven-place logarithms are quite satis- 
factory, for two places of significant figures are unavoidably lost 
in the calculation of tables ; one place is lost in calculating 
the constants, and one is lost in applying the equations, leav- 
ing the fifth place, which should appear in the table, subject 
to error. 

Equations for the Pressure of Steam at Paris. — In 
view of the preceding statements it appeared desirable to re- 
calculate the constants for equations B and C with such a 
degree of accuracy as to exclude any doubt as to the relia- 
bility of the results. Accordingly the logarithms of the five 
values of/ for each equation were taken from Vega's ten-place 
table, and then the remainder of the calculations were carried 
on with natural numbers, checking by independent methods, 
with the following results : 

B. For steam from o° to ioo° C, 

log p — a — ba n -f- c(i n ; 

a = 4.7393622142 ; 
log £ = 0.6117400190; 
log c — 8.1320378383 — 10 ; 
log a= 9.996725532820 — 10 ; 
log ft = 0.006864675924 ; 

n = t. 

C. For steam from ioo° to 220 C, 

log p = a — bot n -f- eft" ; 

« = 5-4574301234 ; 

log b =0.4119787931 ; 
log c = 7.7417476470 - 10 ; 
log a= 9.99741106346 — 10 ; 
log fS = 0.0076424891 13 ; 
n = t — 100. 

Pressure of Steam at Latitude 45 , French System. — 

It is customary to reduce all measurements to the latitude of 
45 and to sea-level. The standard thermometer should 
then have its boiling- and freezing-points determined under, or 



SA TURA TED VAPORS, 87 

reduced to, such conditions. The value of g, the accelera- 
tion due to gravity given by equation (61), is 9. 8092 1 8 metres 
at Paris, latitude 48 50' 14", and at an elevation of 60 

metres. At 45°and at sea-level £"—9.806056; consequently 
760 mm. of mercury at 45 ° latitude give a pressure equal to 

that of 

980.6056 
760 X 980.9218 = 75 9-755 mm. 

at Paris, and by equation B this corresponds to a temperature 

of 99 . 99 1 C. In other words, the thermometer which is 

standard at 45 ° has each degree 0.99991 of the length of, the 

degree of a thermometer standard at Paris. 

Again, the height of a column of mercury at 45 ° latitude 

980.9218 . . 

is -5 — 7 — 7 times the height of a column which will give the 
980.6056 ^ s 

same pressure at Paris. Consequently to reduce equation B 

to 45 latitude we have 

980.9218 
log p = a + log ^ 8q ^ q56 - top-***' + cfi*™* 

and for equation C 

log/ = a -\~ log ';? Y — £of(°-9999i'~ioo) _j_ ^(0.90991/ - 100) 

980.9218 , ,. , 

— a 4- log £<* ~ °-°°9a:o-999i('' - 100) 

n & 980.6056 

1 c ft - 0.009 y£. 9 999j(* - 100). 

The resulting equations are : 

B. For steam from o° to ioo° C. at 45 latitude, 

\ogp = a x -6a x n + cfi*; 

a, — 4.7395022; 
log b = o. 6 1 1 7400 1 90 ; 
log c = 8.1320378383 - 10; 



88 THERMODYNAMICS OF THE STEAM-ENGINE. 

loga l =: 9.996725827522 — 10; 
log /?, = 0.006864058103 ; 
n = t. 

C. For steam from ioo° to 220 C. at 45 ° latitude, 
log/ =a g - b x a? + c x fl*\ 

"x = 5.457570I; 

\ogb x — 0.4120020935; 

log c x — 7.7416788646 — 10; 

log 0L X = 9.9974I I296464 — IO" 

log ft = 0.007641 80 1289; 
n—t— 100. 

Pressure of Steam at Latitude 45 . English System. — 

To reduce the equations for the pressure of steam so that 
they will give the pressures in pounds on the square inch for 
degrees Fahrenheit there are required the comparison of 
measures of length and of weight, the comparison of the scales 
of the thermometers, and the specific gravity of mercury. 

Professor Rogers * gives for the length of the metre 39. 3702 
inches. This differs from the value given by Captain Clarke f 
by an amount that does not affect the values in the tables, 
his value being 39.370432 inches. 

Professor Miller % gives for the weight of one kilogram 
2.20462125 pounds. 

Regnault § gives for the weight of one litre of mercury 
13.5959 kilograms. 

The degree Fahrenheit is -§ of the degree centigrade. 

13-5959 X 2.204621 

Let k = — 3 ; 

39-3702 

* Proceedings of the Am. Acad, of Arts and Sciences, 1882-83; also addi- 
tional observations. 

\ Proceedings of the Royal Society, vol. XV, 1866. 
\ Philosophical Transactions, cxlvi, 1856. 
v^ Memoir es de V Institut de France, vol. xxi. 



SATURATED VAPORS. 89 

then equations B and C have for the reduction to degrees 
Fahrenheit, and pounds on the square inch, 

log p = a x -f- log k — baf n -\- cfi* n ; 

log / = a x + log k - b x a}* + c x fi**. 

The resulting equations are : 

B. For steam from 32 to 212 F. in pounds on the square 
inch, 

\og/> = a, - baf-\-cPf\ 

tf 2 = 3.025908; 
log b = o. 6 1 1 7400 ; 
log c = 8.13204 — 10; 
log a 2 = 9.998181015 — 10; 
log fi t = 0.0038134; 
n z=z t — 32. 

C. For steam from 2 12 to 428 F. in pounds on the squarer 
inch, 

log/ = a % — b x a* + c x fif\ 

",= 3.743976; 
\ogb x = 0.412002 1 ; 
log c x = 7.74168 — 10; 
log", = 9-99 8 56i83i — 10; 
logA = 0.0042454; 
n = t — 212. 

Other Equations for the Pressure of Steam. — A num- 
ber of other forms have been proposed for the empirical 



90 THERMODYNAMICS OF THE STEAM-ENGINE. 

equations for calculating the pressure of saturated steam ; one 
of the best is that given by Rankine * having the form 

log/= ^ __ _ _. ... (107) 



Assuming the absolute zero on the Fahrenheit scale to be 
at — 461 °. 2, he computed for pressure on the square inch the 
following values for his constants : 

A =6.1007; log B = 3-43642; log C= 5-59 8 73. 

This equation has the advantage that it may be solved 
directly for T, a property that Regnault's equations do not 
have. It gives fairly accurate results, and the greater part of 
English tables of properties of saturated steam are calculated 
by its aid. 

A number of exponential formulae have been devised, of 
which the principal advantage is the facility of application. 
The following, by Magnus, gives pressures in mm. of mercury 
for degrees centigrade, and agrees quite well w r ith Regnault's 
results below 100°, but is not so correct above 100°: 



7-4475^ 

/ = 4.525 X 10234.69 + ' (108) 



Pressure of Other Vapors. — Regnault f determined also 
the pressure of a large number of saturated vapors at various 
temperatures, and deduced equations for each in the form of 
equation (94). The equations and the constants as deter- 
mined by him for the commoner vapors are given in the fol- 
lowing table : 



* Steam-engine and Other Prime Movers. 

\ Academic des Sciences, Comptes rendus, tome xxxvi. 



SA T URA TED VA PORS. 
PRESSURE OF SATURATED VAPORS. 



9 1 



Alcohol 

Ether 

Chloroform 

Carbon bisulphide... 
Carbon tetrachloride. 



log p 



a — ba n + cfi n 
a + ba» — cft n 
a — bet* _ c fin 
a — ba n — eft' 1 
a — ba n — cfi n 



5.4562028 
5.0286298 
5.2253893 
5.4011662 
12.0962331 



4.9809960 
0.0002284 
2.9531281 
3.4405663 
9.1375180 



0.0485397 
3.1906390 
0.0668673 
O.2857386 
1.9674890 



Alcohol 

Ether 

Chloroform 

Carbon bisulphide... 
Carbon tetrachloride. 



log a 



1.99708557 
0.0145775 
1. 9974144 
1. 9977628 
1. 9997120 



log£ 



I.9409485 
I.996877 
T.9868176 
I.99II997 
T. 9949780 



t + 20 

t -\- 20 
t — 20 

t + 20 
t + 20 



Limits. 



- 20°, + I 5 0° C. 

- 20°, -f 120° C. 

4- 20 , + 164 C. 

- 20°, + I4O C. 

- 20°,+ 188 C. 



Zeuner* states that there is a slight error in Regnault's cal- 
culation of the constants for aceton, and gives instead 

log p = a — ba n -L. cfi n ; 

a = 5.3085419; 
log bot n = -U- 0.5312766 — 0.0026148/; 
log cfi" = — 0.9645222 — 0.0215592/. 

Differential Coefficient ~. — From the general form of 
the equation (94) we have 



log. p = a -| ba n A cQ n . 



(109) 



M being the modulus of the common system of logarithms. 
Differentiating, 

j| = i*l°*«. -+-1*106, /»./»-; 



* Mechanische War?netheorie. 



92 THERMODYNAMICS OF THE STEAM-ENGINE, 

or, reducing to common logarithms, 

■■■ ii= A «"+ £fi " ^ 

For saturated steam at 45 ° latitude the constants to be 
used with equation (no) are: 

French units. 

B. For o° to ioo° C, mm. of mercury, 

log A = 8.85 12729 — 10; 
log 5 = 6.69305 - 10; 
log a, = 9.996725828 — 10; 
log fi x ~ 0.0068641. 

C. For 100 to 220 C, mm. of mercury, 

log ,4 = 8.5495158- 10; 
log B = 6.34931 - 10; 
loga 1 = 9.99741 1296- 10; 
log/Jj = 0.0076418. 



English units. 

B. For 32 to 212 F., pounds on the square inch, 

log A — 8.5960005 — 10; 
log B = 6.43778 - 10; 
log or, — 9.998181015 — IO* 
log/? 2 = 0.0038134. 

C. For 212 to 428 F., pounds on the square inch, 

log A — 8.2942434— 10; 
log B — 0. 09403 — 10; 
logar a = 9.998561831 — IO; 
log/? 3 = 0.0042454. 



SA TUP A TED VA PORS. 



93 



dp 
It is to be remarked that -f may be found approximately 

at 

by dividing a small difference of pressure by the corresponding 

Ap 



difference of temperature ; that is, by calculating 



At 



With a 



table for even degrees of temperature we may calculate the 
value approximately for a given temperature by dividing the 
difference of the pressures corresponding to the next higher and 
the next lower degrees by two. 

The following table of constants for the several vapors 
named were calculated by Zeuner from the preceding equa- 
tions for temperature and pressure of the same vapors : 



DIFFERENTIAL COEFFICIENT 



i dp 
p~di' 



Alcohol 

Ether , 

Chloroform 

Carbon bisulphide. 
Carbon tetrachloride. . . 
Aceton 



Sign. 


Aa n 


B$ n 










+ 






+ 






f 


+ 


4- 


H 


\- 


+ 



log Ma") 



1.1720041 
1.3396624 
1.3410130 

!• 4339778 
I. 861IO78 
!• 3268535 



O.OO29143/ 
O.OO31223/ 

o. 00258562" 

0.00223722" 
O.OOO2880/ 
O.OO26148/ 



log (2?J3«) 



[.9992701 — o. 0590515/ 
\- 4616396 + 0.0145775/ 
1.0667124 — 0.0131824/ 
5.051 1078 — 0.0088003/ 
:.38i2ig5 — 0.0050220/ 
[.9064582 — 0.0215592/ 



Rowland's Experiments. — The most accurate and re- 
liable determinations of the mechanical equivalent of heat 
were made by Rowland,* who found that there is a notable 
variation of the mechanical equivalent at low temperatures. 
His experiments give a very delicate determination of the 
specific heat of water at low temperatures, and consequently of 
the heat of the liquid, i.e., the heat required to raise water 
from freezing-point to a given temperature. 

The apparatus used was similar to that used by Joule, with 
modifications to give greater certainty of results. The calo- 
rimeter was of larger size, and the paddle had the upper vanes 
curved like the blades of a centrifugal pump, to give a strong 



Proceedings of the American Academy, vol. xv (N. S. vii), 1879. 



94 T HER MOD YNAMICS OF THE STEAM-ENGINE. 

circulation up through the centre, past the thermometer for 
taking the temperatures, and down at the outside. The paddle 
was driven by a petroleum-engine, and the power applied was 
measured by making the calorimeter into a friction-brake, with 
two arms at which the turning moment was measured. Radia- 
tion was made as small as possible, and then was made deter- 
minate by use of a water-jacket outside of the calorimeter. 

The experiments consisted essentially in delivering a meas- 
ured amount of work to the water in the calorimeter and in 
measuring the rise of temperature due to it. The whole 
range of experiments was from 2° to 4i°C. It is, however, 
convenient to make our calculations of the properties of 
saturated steam and of water begin at freezing-point ; conse- 
quently for this purpose it has been assumed that the mechani- 
cal equivalent of heat is constant from freezing-point to 3 C. 
The work required to raise one kilogram of water from 2° to 
3 is 430 metre-kilograms, so that our assumption gives for the 
work required to raise one kilogram of water from freezing- 
point to 3 , 1290 metre-kilograms. Any error that may come 
from this assumption is eliminated in practical calculations, for 
we always deal with differences of the heat of the liquid, and 
seldom or never have a temperature so low as 2° C. With the 
assumption just made Rowland's experiments may be ex- 
pressed by the table on page 95. 

In column 2 is given the work in metre-kilograms required 
to raise one kilogram of water from freezing-point to the tem- 
perature given in column 1, while column 3 gives the most 
probable mechanical equivalent of heat for the several tempera- 
tures from 5 to 36 C. The other two columns will be dis- 
cussed under the head of Specific Heat of Water. 

Standard Temperature. — In the beginning of our work 
it was stated that we should use 62 F. for our standard tem- 
perature; and the reasons for so doing may now be shown. 
We know actually nothing about the specific heat of water 
from o° to 2° C. ; consequently the commonly accepted value 
of the thermal unit — i.e., the heat required to raise one unit 



SATURATED VAPORS. 



95 



ROWLAND'S MECHANICAL EQUIVALENT OF HEAT. 



bo 



9 
10 

ii 

12 

13 
14 
15 
16 

17 
18 

19 

20 
21 



u 




<u 




•o £ 


c 


a u s 


rtjU . 


3 «2 


■-g« 


£ « bfl 


C > <U 


-So 


« 3ffi 


§o5 


4(W O 


H 


S 


2 


3 


430 




860 




I200 


..... 


1721 




2I50 


429.8 


2580 


429-5 


3009 


429-3 


3439 


429.0 


3868 


428.8 


4296 


428.5 


4723 


428.3 


5i5i 


428.1 


5578 


427.9 


6006 


427.7 


6433 


427.4 


6861 


427.2 


7289 


427.0 


7717 


426.8 


8144 


426.6 


8571 


426.4 


8997 


426.2 



**d 


"rt 


<uy « 


<D(J 


•"= ~e 




•^•0 w 


""6V, 






03C 


3 ^ 


*j o*u 


« o*-3 




4J1-J 


PC 


PC 


4 


5 


1.0068 


1 .007 


2.0135 


2.014 


3.0204 


3.022 


4.0295 


4.029 


5-Q339 


5.036 


6 . 0408 


6.040 


7.0452 


7-045 


8.0520 


8.049 


9-0564 


9-054 


10.059 


10.058 


11.058 


1 1 . 060 


12.061 


12.061 


13.060 


13-063 


14.063 


14.064 


15*065 


15.066 


16.004 


16.066 


17.066 


1 7 . 066 


18.068 


18.066 


19.068 


19.066 


20.068 


20.066 


21.065 


2 1 . 064 



pub 

S <L> « 

3 *j u 

-So 

H 
2 



22 
23 
24 
25 
26 

27 

28 

29 
30 
31 
32 

33 
34 
35 
36 
37 
38 

39 

40 

4i 



9424 

9850 

10277 

10701 

11128 

H553 
11978 
12399 
12828 
13253 
13675 
14101 

14527 
14952 

15379 
15805 
16231 
16657 
17083 
17508 



— c 

C > (U 



u 



>< 




«H 


n! 


^T3 


C 


O 3 


y 


_ cr 


u 


a-: 

4JI-I 


a 


PC 




4 





426.1 


22.065 


426.0 


23.063 


425-9 


24.062 


425.8 


25.055 


425.7 


26.054 


425.6 


27.050 


425-6 


28.045 


425-5 


29.031 


425.6 


30.035 


425.6 


31.030 


425.6 


32.018 


425-7 


33.0I6 


425-7 


34.011 


425.8 


35.008 


425.8 


36 . 008 




37.007 




38.003 




39 . 000 




39.998 




40.993 






= H 

*, criS 
rt-; 3 



22.063 
23.061 

24.059 
25.058 
26.O53 
27.O48 
28.O42 
29.O37 
3O.O32 
3I.027 
32.023 
33.0I8 
34.OI4 
35.OO9 
36 • OO7 
37.005 
38 . OO4 
39.002 
40 . OOO 



of weight of water from o° to i° C, or from 32 to 33 F. — is 
an ideal quantity inferred from the behavior of water at higher 
temperatures. It is more scientific to take an easily verified 
quantity for the standard; and there is a practical convenience 
in choosing 62 ° F. for the standard temperature, because it is 
near the mean temperature of the air during experimental 
work. Therefore it is near the mean temperature in the 
calorimeter during ordinary work with that instrument; and 
the specific heat of water for the range of temperature in the 
calorimeter may usually be considered to be unity, without 
error, unless great refinement is desired. Moreover, 62 F. 
is the temperature at which the English units of weight and 
measure are standard. 

Mechanical Equivalent of Heat. — 62 F. corresponds 
with i6f° C, at which the mechanical equivalent of heat given 



9 6 



THERMODYNAMICS OF THE STEAM-ENGINE. 



in the table of Rowland's experiments is 427.1. The value 
of g at Baltimore, latitude 39 17', is 980.05 centimetres; 
therefore, reducing to 45 of latitude, where g= 980.6056 
centimetres, the value of./ is 

980.05 
/= 427.1 X 8q 6q 6 = 426.9 metre-kilograms. 

To reduce to the English system of units it is sufficient to 
multiply by -§-, giving 

/= 778 foot-pounds. 

Since the value given by Joule is commonly quoted, it 
will be of interest to make a comparison of his latest work 
(1873) with Rowland's, as given in the following table: 

COMPARISON OF ROWLAND'S AND JOULE'S EXPERIMENTS. 



Temperature. 


Joule's Value at 

Manchester, 
English System. 


Reduced to the Air-thermometer 
and to the Latitude of Baltimore. 


Rowland's Value 
Corresponding. 


English. 


French. 


12°. 7 
14 -5 

14 -7 

15 -5 
17 -3 


774-6 
767.O 

772.7 

773-1 
774.0 


778.5 
770.5 
776. T 

776.4 
777.0 


427.I 
422.7 
425-8 
426.O 
426.3 


428.O 
427.6 
427-5 
427.3 
426.9 



Specific Heat of Water. — From freezing-point to 40 C. 
the specific heat of water may be determined from Rowland's 
experiments on the mechanical equivalent of heat as arranged 
in the table on page 95. For this purpose we may first divide 
the metre-kilograms required to raise one kilogram of water 
from freezing-point to a given temperature, as set down in 
column 2 of that table, by the mechanical equivalent of heat 
(427.1 kilograms) at the latitude of Baltimore. This gives 
the experimental values of the heat of the liquid given in 
column 4. This column shows some accidental experimental 
irregularities, which were eliminated by drawing a diagram 



SA T URA TED VA PORS. 



97 



with temperatures for abscissae and with heats of the liquid 
for ordinates. That diagram also made it possible to assign 
the series of values of the specific heat of water from freezing- 
point to 40 C. given in the following table: 



SPECIFIC HEAT OF WATER. 



Range. 


Specific Heat. 


Centigrade. 


Fahrenheit. 


O to 5 
5 to 10 
10 to 15 
15 to 20 
20 to 25 
25 to 30 
30 to 35 
35 to 40 
40 to 45 
45 to 155 
155 to 200 


32 to 41 

41 to 50 

50 to 59 

59 to 68 

68 to 77 

77 to 86 

86 to 95 

95 to 104 

104 to 113 

113 to 311 

311 to 392 


I .0072 
I . 0044 
I.OO16 
I. 

O.9984 
O . 9948 

0.9954 
O.9982 
I. 

I.008 
I .046 



The calculated heats of the liquid given in column 5 of the 
table on page 95 were calculated with the specific heats in 
the above table. They are more regular than the quantities 
in the 4th column, and differ from them no more than can 
properly be attributed to accidental irregularities. 

For the heat of the liquid and the specific heat of water 
beyond 40 C. it is necessary to go to Regnault's determina- 
tions. Having satisfied himself that the specific heat of water 
at low temperatures was practically constant, Regnault pro- 
ceeded to determine the specific heat of water at high tem- 
peratures by the method of mixtures, running hot water from 
the water-space of a boiler into a calorimeter containing cold 
water. The record of his work gives forty such tests. To 
prepare these tests for .our purposes it was necessary to cor- 
rect his calculations for the true specific heat of the water in 
the calorimeter; after this was done the true heat of the liquid 
was readily determined and was plotted on the same diagram 
with the heats of the liquid from Rowland's work, and 



98 THERMODYNAMICS OF THE STEAM-ENGINE. 

specific heats for water from 40 to 45 , from 45 to 15 5 , and 
from 1 55 to 200 , centigrade, were assigned as given in the 
above table of the specific heats of water. 

It is interesting to know in this connection that after his 
investigation of the mechanical equivalent of heat and the 
consequent determination of the specific heat of water 
Rowland repeated Regnault's experiments on the specific 
heat of water at low temperatures by the method of mixtures, 
and that he found it possible to recognize the very peculiar 
behavior of water at low temperatures, which is so clearly 
shown by his experiments recorded in the table on page 95 ; 
but his work showed that the unavoidable irregularities of the 
method of mixtures were a large part of the differences 
between the true specific heat and unity, which shows why 
Regnault failed to find any variation in the specific heat of 
water at low temperatures. 

In using the specific heats of water for calculating the heat 
of the liquid by aid of the table on page 97 it is necessary 
to proceed step by step. Thus the heat of the liquid at 
23 C. is 

5 X 1.0072 + 5X 1.0044+5 X 1.0016-f 5 X 1 + 3 X 0.9984 

= 23.061 B. T. U. 

Specific Heat of Other Liquids. — Regnault determined 
the specific heats and the heats of the liquid of various other 
liquids besides those for water, using the method of mixtures 
for all his work. The following table gives his results un- 
altered : 

HEAT OF THE LIQUID. 

Alcohol q = 0.54754/ + 0.0011218/* 

-J- o. 000002 206/ 3 

Ether q = 0.5^901/ -f 0.0002959/ 2 

Chloroform q — 0.23235/ + 0.0000507/ 2 

Carbon bisulphide q — 0.23523/ + 0.0000815/ 2 

Carbon tetrachloride q = o. 19798/ -+- 0.0000906/ 2 

Aceton q = o. 50643/ -f 0.0003965/ 2 

Water q — t + 0.00002/' -f- 0.0000003/ 3 



SA TUKA TED VAPORS. 99 

Regnault's equation for water is included here to make 
his work complete; it of course gives very different results 
from those by the table of specific heats on page 97. 

The specific heat for any liquid may be determined at any 
temperature by differentiating the corresponding equation for 
the heat of the liquid. Thus Regnault's equation for the 
specific heat of water is 

da 
c = -j = I -f- 0.00004/ + 0.0000009/'. 

Total Heat. — This term is defined as the heat required to 
raise a unit of weight of water from freezing-point to a given 
temperature, and to entirely evaporate it at that temperature. 
The experiments made by Regnault were in the reverse order; 
that is, steam was led from a boiler into the calorimeter and 
there condensed. Knowing the initial and final weights of 
the calorimeter, the temperature of the steam, and the initial 
and final temperatures of the water in the calorimeter, he was 
able, after applying the necessary corrections, to calculate the 
total heats for the several experiments. 

As a conclusion of the work he gives the following values 
for the total heats: 

io° 610 By equation, 609.6 

63 625 625.2 

100 ~ 637 

195 666 

Assuming an equation of the form 

\ = A+Bt, (m) 

Regnault calculated the constants from the values given for 
100° and 195 , and gives the equation 

X = 606.5 + 0.305/ (112) 



100 THERM 0D YNAMICS OF THE STEAM-ENGINE. 

An investigation of the original experimental results, 
allowing for the true specific heat of the water in the calo- 
rimeter, showed that the probable errors of the method of 
determining the total heat were larger than the deviations 
of the true specific heats from unity, the value assumed by 
Regnault; and, further, it appeared that his equation repre- 
sents our best knowledge of the total heat of steam. The 
probable error appears to lie between T oVo an ^ jwo, making 
the total heat the most uncertain of the experimental proper- 
ties of steam. 

For the Fahrenheit scale the equation becomes 

1 = 1091.7-fo.305i/- 32). . . . (113) 

Regnault gives the equations following for other liquids: 

Ether A = 94 -f 0.45* — 0.0005 55 56* 2 

Chloroform A = 67 +0.1375* 

Carbon bisulphide. .. A = 90 -f 0.14601* — 0.0004123* 2 

Carbon tetrachloride A = 52 + 0.14625* — 0.000172* 2 

Aceton A = 140.5 + 0.36644* — 0.000516* 2 

Heat of Vaporization. — If the heat of the liquid be sub- 
tracted from the total heat, the remainder is called the heat 
of vaporization, and is represented by r, so that 

' = A. — q (114) 

Specific Volume of Liquids. — The coefficient of expan- 
sion of most liquids is large as compared with that of solids, 
but it is small as compared with that of gases or vapors. 
Again, the specific volume of a vapor is large compared with 
that of the liquid from which it is formed. Consequently the 
error of neglecting the increase of volume of a liquid with the 
rise of temperature is small in equations relating to the ther- 
modynamics of a saturated vapor, or of a mixture of a liquid 
and its vapor when a considerable part by weight of the mix- 
ture is vapor. It is therefore customary to consider the 
specific volume of a liquid a to be constant. 



SA TURA TED VAPORS. 



101 



The following table gives the specific gravities and specific 
volumes of liquids: 

SPECIFIC GRAVITIES AND SPECIFIC VOLUMES OF LIQUIDS. 



Alcohol 

Ether 

Chloroform 

Carbon bisulphide . . 
Carbon tetrachloride 

Aceton 

Sulphur dioxide 

Ammonia 

Water 



Specific 

Gravity, 

compared 

with Water 

at 4 ° C 



O.80625 
O.736 

1-527 

I.2g22 

I.6320 

O.81 

1-4336 

O.6364 



Specific Volume. 



Cubic Metres. Cubic Feet 



O.OO1240 

O.OOI350 

O.OO0655 

O.OOO774 

O.O0613 

O.OOI23 

O.OOO6981 

O.OOI571 

O.OOI 



O.OII2 
0.0252 
O.OI602 



Experiments were made by Hirn * to determine the 
volumes of liquid at high temperatures compared with the 
volume at freezing-point, by a method which was essentially 
to use them for the expansive substance of a thermometer. 
The results are given in the following equations: 
SPECIFIC VOLUMES OF HOT LIQUIDS. 



Water, 

TOO° C. to 200° C. 

(Vol. at 4 = unity.) 



Alcohol, 

30 C. to 160 C. 

(Vol. at o D = unity.) 



Ether, 

30 C. to 130 C. 

(Vol. at o° = unity.) 



Carbon Bisulphide, 
30° C. to 160 C. 

(Vol. at o" = unity.) 



Carbon Tetrachloride, 

30 C. to 160 C. 

(Vol. at o° = unity.) 



v = 1 + 0.00010867875* 

-J- o. 000003007365 3* 2 
-j- 0.00000002S730422* 3 



Logarithms. 

6.0361445 — IO 
4.4781862 — IO 
1. 4583419 — IO 



O.OOOOOOOOOO06645 703 1/ 4 8.8225409 — 20 



v = I -j- O.OOO73892265* 

+ 0.00001055235^ 

— 0.000000092480842^ 
-\- 0.0000000004041 3567^ 

v = I + O.OO13489059* 
-j- 0.0000065537* 2 

— o. 0000000344907 56* 3 
-f- 0.00000000033772062* 4 

v — 1 -f- 0.0011680559* 

-j- 0.0000016489598^ 

— 0.0000000008 1 1 1906 2 t z 
-J- 0.0000000000609465 89** 

v = 1 -f- 0.0010671883* 

-j- o. 000003565 1378Z 2 

— 0.0000000149492 8 1* 3 

-f- 0.000000000085 1 823 1 8Z 4 



6.8685991 — ro 
3.0233492 — 10 
2.9660517 — 10 
0.6065278 — 10 

7.1299817 — 10 
4.8164866 — 10 
2.5377028 — 10 
0.5285571 — 10 

7.0674636 — 10 
4.2172103 — 10 
0.9091229 — 10 
9.7849494 — 20 

7.0282409 — 10 
4.5520763 — 10 
2.1746202 — 10 
9.9303494 — 20 



* Annates de Chimie et de Physique, 1867. 



102 THERMODYNAMICS OF THE STEAM-ENGINE. 

Internal and External Latent Heat. — The heat of 
vaporization overcomes external pressure, and changes the 
state from liquid to vapor at constant temperature and pres- 
sure. Let the specific volume of the saturated vapor be s 
and that of the liquid be cr; then the change of volume is 
s — cr = u on passing from the liquid to the vaporous state. 
The external work is 

p{s — o)=pu, . . . . . (115) 

and the corresponding amount of heat, or the external latent 
heat, is 

Ap(s — cr) = Apu (116) 

The heat required to do the disgregation work, or the 
internal latent heat, is 

p—r — Apu. .... . (117) 

General Equation. — A pound or a kilogram of a mixture 
of a liquid and its vapor consists of a certain part, x, of 
vapor, and the remainder, 1 — x, of liquid. The specific 
volume of the mixture is 

v—xs-\-{i — x)cr = (s — cr)x -\- cr = ux -f- cr, (1 18} 

in which s is the specific volume of saturated vapor, cr is the 
specific volume of the liquid, and u is the increase of volume 
due to vaporization. 

Since the pressure of saturated vapor depends on the tem- 
perature only, the variables in the general equation for a mix- 
ture of a liquid and its vapor are temperature and volume; 
and the specific volume as shown by equation (118) depends 
upon x, the condition of the mixture, and on s or u, which in 
turn depend on the temperature only. Consequently the 
general equation may be expressed as a function of t and x. 

When a mixture of liquid and its vapor receives heat 
there is in general an increase in the temperature of the por- 
tion x of vapor and in the portion 1 — x of liquid, and there 



SATURATED VAPORS. 



I03 



is a vaporization of part of the liquid. Taking c for the 
specific heat of the liquid and h for the specific heat of the 
vapor, while r is the heat of vaporization, we shall have for 
an infinitesimal change 



dQ = hxdt -j- c{ 1 — x)dt -\- rdx. 



(119) 



Application of the First Law. — The first law of ther- 
modynamics is applied to equation (1 19) by combining it with 
equation (23), so that 

dQ = A(dE -\-pdv) = hxdt -\- c(\ — x)dt -f- rdx ; (120) 



1 r 

dE = ~j[hx -\- c{ 1 — x)~]dt -\- —rdx — pdv. 



A 



A 



(121) 



But 



+ 



dE 



r 
A 



pi \dx)\ dx > 



(122) 



Since 



d*E 

dx dt 



d*E 

dt dx' 



i\jv*+<i-*)-} 



p\ 



[dv 



\dt 



d L 
dt 



r (dv\ 

-p\- 



A 



\dxft_ 



Bearing in mind that h, c, and / are functions of t, and 



not of x, the differentiation gives 



A 



1 • d 2 v 

{h-c)-p- 



dx dt 



1 dr 
A~dt 



dp(dv\ d*v 

dt\dx)t~ P ~dUbc' ( I23 ) 



Since/ and r are functions of t, and not of x, the expres- 

dr dp . . idr\ idp\ 

sions -77 and -7- are written instead of -r and -r . From 

dt dt \dtjx \dtjx 

equation (1 18) we have (& being constant) 

'dv\ 

dx) t = "' 



104 THERMODYNAMICS OF THE STEAM-ENGINE. 

and, further, we have 

d*v d 2 v 



dx dt dt dx' 



so that equation (123) reduces to 



dr 



dp 



dt+ C ~ k = Au dt 



(124) 



Application of the Second Law. — By this law -=■ is an 
exact differential. 



dQ hx 4- c( 1 — x) T r 

-ldt+ Y dx. . . (125) 



T 



T 



But 



d*<f> W _ J T a J T 



dx dt dt dx 




dx dt dt dx 


d 


hx 


+ <i- 


*)i 


< dt\T) x ' ' ' 


dx 




T 






h — c 
T ~ 


rr.dr 

T Jt~ r 




y-«2 > 


, 




dr 
dt~ 


- k: 


r 
' T 



(126) 



(127) 



First and Second Laws Combined. — The combination 
of equations (124) and (127) gives 



r = AuT 



dp 
It' 



. (128) 



SATURATED VAPORS. • 105 

Natures of the Specific Heats. — Both the specific 
pressure p and the specific volume s of saturated vapor 
depend on the temperature, so that we can have neither 
specific heat at constant volume nor specific heat at constant 
pressure, as we had with perfect gases. The specific heat c 
for the liquid and the specific heat h for the vapor are the 
amounts of heat required to raise the temperature of one unit 
of weight one degree, under the condition that the pressure 
shall rise with the temperature, according to the law for 
saturated vapor. The volume of the liquid, indeed, changes 
so slowly that we can ignore it; but the volume of the vapor 
changes rapidly. The specific heat of the liquid as determined 
by Rowland up to 40 C. was at atmospheric pressure, but 
Regnault's determinations for higher temperatures were under 
a varying pressure; for our present purpose we may assume 
the determinations by both experimenters to conform to our 
definition just given. 

Equation (127) gives a ready way of calculating the 
specific heat for a vapor, for from it 

, dr r 
/l = c +Vt-T < 12 9) 

Now r may be readily expressed as a function of t y and then 

dr 
by differentiation -=■ may be determined. For steam 

r = X- q — 606.5 4- 0.305/ + ^ + *(*— t,), 

in which t l is the temperature at the beginning of the range, 
as given by the table on page 97, within which t may fall. 
Therefore 

dr 



and 



s = 0.305-*, 



h = 0.305 — -^. .... (130) 



106 THERM OD YNAMICS OF THE STEAM-ENGINE. 

For other vapors the equations, deduced from the empirical 
equations for X and q on pages 98 and 100, are somewhat 
more complicated, but they involve no especial difficulty. 

The following table gives the values of h for steam at 
several absolute pressures: 

SPECIFIC HEAT OF STEAM. 

Pressures, lbs. per sq. in.,/.. 5 50 100 200 300 

Temperatures, t° F 162.3 280.9 327.6 381.7 417.4 

Specific heat, h —1.607 — 1.237 — 1.122 — 1.001 — o.g3r 

The negative sign shows that heat must be abstracted from 
saturated steam when the temperature and pressure are 
increased, otherwise it will become superheated. On the 
other hand, steam, when it suddenly expands with a loss of 
temperature and pressure, suffers condensation, and the heat 
thus liberated supplies that required by the uncondensed 
portion. 

Hirn * verified this conclusion by suddenly expanding 
steam in a cylinder with glass sides, whereupon the clear 
saturated steam suffered partial condensation, as indicated by- 
the formation of a cloud of mist. The reverse of this experi- 
ment showed that steam does not condense with sudden com- 
pression, as shown by Cazin. 

Ether has a positive value for h. As the theory indicates, 
a cloud is formed during sudden compression, but not duiing 
sudden expansion. 

The table of values of h for steam shows a notable decrease 
for higher temperatures, which indicates a point of inversion 
at which h is zero and above which h is positive, but the 
temperature of that point cannot be determined from our 
experimental knowledge. For chloroform the point of inver- 
sion was calculated by Cazin f to be I23°.48, and determined 
experimentally by him to be between 12 5 and 129 . The 
discrepancy is mostly due to the imperfection of the apparatus 

* Bulletin de la Socie'te Industr. de Mulhouse, cxxxiii. 
\ Comptes rendus de V Acade'mie des Sciences , lxii. 



SA T URA TED VA FORS. I O/ 

used, which substituted finite changes of considerable magni- 
tude for the indefinitely small changes required by the theory. 
Specific Volume and Density. — The most important 
result of the application of the methods of thermodynamics 
to the properties of saturated vapor is expressed by equation 
(128), which gives a method of calculating the specific volume; 
thus: 

dt 

The numerical value of cr for water for French units is 
O.ooi, and for English units is ~ = 0.016, nearly. The 
density, or weight of a unit of volume, is of course the 
reciprocal of the specific volume. 

It is of interest to consider the degree of accuracy that 
may be expected from this method of calculating the density 
of saturated vapor. The value of r depends on A and q\ for 
the first Regnault gives three figures in the data from which 
the empirical equation is deduced, and the experimental work 
does not indicate a greater degree of accuracy. The fourth 
figure, if stated, is likely to be in error to the extent of five 
units. The value of T is commonly stated in four figures, of 
which the last may be in error by two units. A as deter- 
mined by Rowland has four figures, the last being uncertain 
to the extent of one or two units. The differential coefficient 

dp 

-j is deduced from the equations for calculating/; and those 

equations are derived from data having five places of signifi- 
cant figures. Now each of the equations B and C, for steam 
at 45 latitude for the English system, gives a pressure of 
14.6967 pounds on the square inch; but the specific volume 
calculated by aid of equation B is 26.550 cubic feet, while 
equation C gives 26.637 cubic feet. The mean, 26.60, differs 
from either extreme by about one in seven hundred. This 
discrepancy is due to the fact that the curves represented by 



108 T HER MOD YNAMICS OF THE STEAM-ENGINE. 

equations B and C meet at the common temperature, 2 12°, 
but do not have a common tangent. Since the equations are 
empirical and not logical, the error or uncertainty is unavoid- 
able, and all calculated specific volumes are affected by a 
similar uncertainty. The greatest probable error is in deter- 
mining r, for which it may be about one in one thousand. 
The error introduced into this equation by using the values of 
A in common use, that is, 772 instead of 778, is about one in 
one hundred. 

In all recent tables of the properties of saturated vapor 
the specific volumes are calculated by the method just dis- 
cussed, on account of the great difficulty of experimental 
determinations. The error of the calculation is not greater 
than the errors of some other parts of the table which are 
determined by direct experiments. 

Experimental Determinations of Specific Volume. — 
The uncertainty of the direct determination of the density 
of saturated vapor is due to the difficulty of determining when 
it is dry and saturated; a small quantity of liquid present or 
a slight degree of superheating will introduce serious errors. 

Two series of tests of the specific volume of saturated 
vapors have been made, by Tate and Fairbairn * and by 
Perot. f Though made with much skill and ingenuity, the 
experiments by Tate and Fairbairn are affected by so large 
experimental errors that they would have very little interest 
were it not that some tables of the properties of saturated 
steam now in use depend partly on these tests. The highest 
pressure used for these tests was 114 inches of mercury, or 
about 50 pounds absolute per square inch. Within this 
range the error of Tate and Fairbairn's empirical equation 
deduced from their experiments is three or four per cent; 
beyond fifty pounds the errors become very large. The 
empirical equation just referred to is 



* Philosophical Transactions, vol. cl, i860. 

\ Journal Franklin Institute, vol. cxxxiii, p. 55. 



SA TURA TED VAPORS. 



109 



r= 25.62 + 



49513 

P+0.72' 



where V is the volume of steam compared with that of the 
water from which it is produced, and P is the pressure in 
inches of mercury. 

The principal difficulty in making direct determinations of 
specific volumes of saturated vapor is to be sure that the 
vapor is neither moist nor superheated. For this purpose 
Perot enclosed a glass globe for weighing the vapor in a 
strong receptacle together with a sealed tube of the liquid to 
be tested. Both the receptacle and the globe were exhausted 
by an air-pump, then the temperature was raised and the tube 
of liquid was broken, and the receptacle and the ^lobe were 
filled with vapor. The globe was supported on a frame, so 
as not to be directly in contact with the walls of the recepta- 

PEROT'S EXPERIMENTS ON SPECIFIC VOLUMES OF 
SATURATED VAPOR. 



Name of Vapor. 


Temperatures. 


Specific Volumes, Cubic Metres. 


Experimental. 


By Equation (131). 


Difference. 




68.20 
88.60 
98.IO 
99.60 
ior.50 
124.IO 


5-747 
2-531 
1.782 

1.657 
I-583 
0.766 


5-428 
2.469 
I.768 
I.683 

I-583 
O.7804 


O.329 
O.062 
0.014 

— O.026 
O.O 

— O.14 


Bisulphide of 


84.60 


0.1163 


O.I2I 


— 0.005 


Ether 


28.40 
30.00 
31.70 
31.90 
57-90 
85-50 
IIO.50 


0.4262 
. 4000 

o.375i 

0.3730 

0.1680 

0.07777 

0.04394 


O.429 

O.4OI3 

O.382 

O.380 

O.169 

O.082 

0.0454 


— 0.003 

— O.OOI3 

— 0.007 

— 0.007 

— O.OOI 

— 0004 

— 0.0015 



cle, and being filled with vapor and surrounded by vapor at 
the same pressure it is fair to conclude that it was at the 



HO THERMODYNAMICS OF THE STEAM-ENGINE. 

temperature due to the pressure of the vapor and that it con- 
tained only dry saturated vapor. In preparation for the 
experiment the mouth of the globe was drawn down to a fine 
tube, around which a platinum wire was wound, so that at the 
proper time the opening could be closed by passing an electric 
current through the wire. After the globe was sealed full of 
dry saturated vapor the receptacle was opened and the globe 
taken out and weighed. 

The results of these tests are shown in the accompanying 
table, together with values calculated by aid of equation (131), 

Zeuner's Equation for Internal Heat. — The following 
empirical equations were proposed by Zeuner for calculating 
the heat equivalent of the disgregation work during vaporiza- 
tion. They are most interesting from the light they throw 
on the critical temperature. 



INTERNAL LATENT HEAT. 
French Units. 

Water p = 575-40 — 0.791/ 

Ether p = 86.54 — 0.10648/ — 0.0007160/' 

Chloroform p = 62.44 — 0.11282/ — 0.0000140/ 2 

Carbon bisulphide p = 82.79 — 0.11446/ — 0.0004020/ 2 

Carbon tetrachloride p = 48.57 — 0.06844/ — 0.0002080/ 2 

Aceton p = 131.63 — 0.20184/ — 0.0006280/ 2 

The following table shows that the equation for water 
gives a fair degree of approximation: 

0° 50° 100° I5O 200° 

By equation (117) 575.5 536.3 496.4 457.4 417.4 

By empirical equation. . . 575.4 535-9 496.3 456.8 417. 1 

Critical Temperature. — The empirical equation for 
steam, and also the value of p in the table above by the exact 
method, show that the internal latent heat decreases as the 
temperature rises, and at sufficiently high temperatures it will 
approach zero. If p is made zero in Zeuner's equation for 



SA TURA TED VAPORS. 1 1 1 

steam the corresponding temperature is 720 C, which indi- 
cates that the true point is much beyond the limits of experi- 
ments. 

The temperature at which ft becomes zero for any vapor 
is called the critical temperature, for at that temperature the 
distinction between the liquid and its vapor vanishes, and 
above that temperature the vapor or gas cannot be liquefied 
by pressure alone. It has been proposed to call a substance 
which is above the critical temperature a gas, and one which 
is below a vapor. 

Experiments on liquids strongly heated in strong glass 
tubes show that vaporization proceeds gradually as the tem- 
perature rises, until a temperature is reached at which the line 
of demarcation between the liquid and its vapor becomes 
indistinct. Above that temperature the liquid all disappears, 
and the tube is full of gas. This is the critical temperature. 
Avenarius* by this method determined the critical tempera- 
lure of four liquids. He also selected from Regnault's 
experiments the data most applicable, and from them deduced 
equations like those given by Zeuner for the internal latent 
heat of vapors, and calculated the critical temperature by 
their aid. The results are as follows: 

Experimental. Calculated. 

Ether ig6°.2 C. I96°.8 C. 

Carbon bisulphide 2y6°.i 274°.o 

Carbon tetrachloride 2g2°.$ 298 '.7 

Aceton 246 . 1 230°. 4 

Curve of Constant Steam Weight. — It was formerly 
assumed in the theory of the steam-engine that the inter- 
change of heat between the steam and the iron of the cylinder 
was by radiation; and, further, that the condensation accom- 
panying adiabatic expansion formed a cloud which instigated 
a rapid interchange of heat where before little had occurred. 
The steam-jacket was assumed to impart just heat enough to 

* Poggendorff's Annalen, cli, 1874. 



112 THERMODYNAMICS OF THE STEAM-ENGINE. 

dissipate this cloud and keep the steam dry. Hence the 
curve of dry saturated steam was considered to be of great 
importance in the theory of the steam-engine, and it is some- 
times drawn on indicator-cards instead of the hyperbola. 
The substitution has no good reason, for the curve is not a 
better approximation to the curve drawn by an indicator, and 
is more troublesome to construct. 

The action of steam in the engine-cylinder has been 
proved to be quite different, for an interchange of heat is 
caused by condensation by contact of the steam with the iron, 
or by evaporation of moisture from it, and the curve of 
saturated steam no longer plays an important part in the 
theory of the steam-engine. Still it is of importance as form- 
ing the boundary-line between superheated steam and wet 
steam. 

The curve may be represented very closely by the ex- 
ponential formula 

pv n — p x v" = const 03 2 ) 

Rankine proposed the value -f J for the exponent n, and 
Zeuner has found that 1.0646 gives still a closer approxima- 
tion. The actual curve may be drawn by plotting pressures 
and volumes from a table of the properties of saturated steam. 

Isothermal Lines. — Since the pressure of saturated vapor 
is a function of the temperature only, the isothermal line of a 
mixture of a liquid and its vapor is a line of equal pressures, 
parallel to the axis of volumes. Steam expanding from the 
boiler into the cylinder of an engine follows such a line; that 
is, the steam-line of an automatic cut-off engine with ample 
ports is nearly parallel to the atmospheric line. 

The heat required for an increase of volume at constant 
pressure is 

Q = r(x, — x,), 

which may be obtained by integrating equation (1 19) with the 
assumption that the temperature is constant; or it may be 



SA TURA TED VA PORS. 1 1 3 

written directly, since r is the heat of vaporization, and 
jr a — x x is the weight of liquid vaporized. 

The work done by the vapor during such an expansion is 

W = p(v,-v 1 )=pu(x,-x 1 ). , . . (133) 

Isodynamic or Isoenergic Lines. — The following method 
of treating the isodynamic changes of a mixture of a liquid 
and its vapor gives the solution of all problems that arise, 
although it does not give an equation to the curve of 
pressures and volumes. 

The increase of intrinsic energy of the mixture of a liquid 
and its vapor, above freezing-point, is 

^ = ^(? + *p); (134) 

where q and xp are the heat-equivalents of the vibration and 
disgregation works. The change of intrinsic energy in passing 
from one condition to another is 

-£, -£i = ^fe- ?i + *,P, — *,A). • • (135) 

When the change is isodynamic, the energy remains the 
same by definition, and 

<1* — <lx + **P* — *iPi = o; . . . (136) 

which equation, together with the formulae 

v 9 = x i u a + <r i v x = x t u x + <r, . . . (137) 

gives the means of solving all problems. 

For example, if a mixture of T 9 ¥ steam and y 1 -^ water 
expands isoenergically from 100 pounds absolute to 15 pounds 
absolute the final condition will be 

°i — ^a + ^iPi 2 97-9— 181.8 + 0.9X802.8 

£, 892.6 ^O^D 



114 THERMODYNAMICS OF THE STEAM-ENGINE. 

The initial and final specific volumes are 

V x = X X U X -\- <T = 0.9(4.403 — O.OI60) -f- O.OI60 = 3.964; 
V, — XJi % + <T = 0.9395(26.15 — O.OI6) -f O.OI6 = 24.54. 

The converse problem requiring the pressure corresponding 
to a given volume cannot be solved directly. The only 
method of solving such a problem is to assume a probable final 
pressure and find the corresponding volume ; then, if necessary, 
assume a new final pressure larger or smaller as may be 
required, and solve for the volume again; and so on until the 
desired degree of accuracy is obtained. 

The isoenergic line can be well represented by an expo- 
nential equation, for which the exponent can be determined 
by the method given on page 69. This is very fortunate, as 
there is no ready way of calculating the external work by the 
aid of the usual tables of the properties of saturated steam. 
Having given or determined the initial and final volumes, 
the exponential equation may be determined, and then the 
external work may be calculated by the equation 

'-/**-&-,{'-<&'}■ ■ <■■•> 

For example, the exponent for the equation representing 
the expansion of the problem on page 113 is 

log A — log A log 100 — log 15 

fl = . : = : : 7- = I. O4I, 

log v t — log v x log 24.54 — log 3.964 

and the external work of expansion is 

100 X 144 X 3-9 6 4( /3-9 6 4\ a ° 41 ) ,. 1u 

W— ±*s j j _ \±-Z — L_ ioooooft.-lbs„ 

1. 041 — 1 ( \24. 54/ ) 

Since there is no change in the intrinsic energy during an 
isoenergic expansion, the external work is equivalent to the 



SA TURA TED VAPORS. I I 5 

heat applied. Thus in the example just solved the heat 
applied is equal to 

IOOOOO -r- 778 = I29 B. T. U. 

Entropy of the Liquid. — Suppose that a unit of weight 
of a liquid is intimately mingled with its vapor, so that its 
temperature is always the same as that of the vapor; then if 
the pressure of the vapor is increased the liquid will be 
heated, and if the vapor expands the liquid will be cooled. 
So far as the unit of weight of the liquid under consideration 
is concerned the processes are reversible, for it will always be 
at the temperature of the substance from which it receives or 
to which it imparts heat, i.e., it is always at the temperature 
of its vapor. 

The change of entropy of the liquid can therefore be cal- 
culated by equation (36), 

J* dQ 
d(f> = -jTj 

which may here be written 

'■=/? = /¥ C39) 

Now the specific heat of water as given in the table on 
page 97 is constant within certain ranges and varies from one 
range to the next range of temperature. The calculation, like 
that for the heat of the liquid, must be made step by step. 

For example, the increase of entropy of water from freez- 
ing-point to 1 3 C. is 

T T T 

1.0072 \og e -^- + 1.0044 log, 7^-°+ 1.0016 log, y- 3 = O.04663. 

-* -* 5 ■*■ 10 

For a liquid like ether which has the heat of the liquid 
represented by an empirical equation, 

q = 0.52901/ -j- 0.0002959/', 



Il6 THERMODYNAMICS OF THE STEAM-ENGINE. 

the specific heat is first obtained by differentiation, giving 

c — 0.52901 + 0.0005918^. 

Then the increase of entropy above that for the freezing-point 
of water may be obtained by aid of equation (139); which 
gives for ether with the French system of units 



6= C j 0.52901 +0.0005918(7^- 273.7) j —; 
^273.7 ' ) ■*■ 

.'.O^z J (0.3670^ + 0.0005918^) ; 

T 

.-. 6 = 0.0005918(7^- 273.7) + 0.3670 log,-——; 

z / 5'7 

T 
.-. 6= 0.0005 9 1 ** + 0.3670 log, .— -. . . . (140) 

For temperatures below the freezing-point of water equa- 
tion (140) gives negative numerical results. 

Other liquids for which equations for the heat of the liquid- 
are given on page 98, may be treated in a similar method. 

Tables of the properties of saturated vapor should include 
entropies of the liquid, calculated from freezing-point, by one 
of the methods just illustrated. If such tables are not at hand, 
then changes in the entropy of the liquid may be calculated 
approximately, on the assumption that the specific heat is a 
constant by the equation 



6 



*dt . Z 



" ° l ~ C J T = C loge ^f' * * ' ( I41 ) 



Entropy due to Vaporization. — When a unit of weight 
of a liquid is vaporized r thermal units, equal to the heat of 
vaporization, must be applied at constant temperature. If 
only the portion x is vaporized, then xr thermal units are 
applied. Treating such a vaporization as a reversible process, 
the change of entropy may be calculated by the equation 







*dQ 1 /» xr 



- <p i = f Y = -^f d Q=Y' - - ( I42 > 



SA TURA TED VAPORS. 1 1 7 

Entropy of a Mixture of a Liquid and its Vapor. — The 

increase in entropy due to heating a unit of weight of a liquid 
from freezing-point to the temperature t and then vaporizing 
x portion of it is 

xr 

e + y, (143) 

where 6 is the entropy of the liquid and r is the heat of 
vaporization, both of which are given in tables of the proper- 
ties of vapors; while T is obtained by adding the absolute 
temperature of zero to the temperature by the thermometer. 
For any other state determined by x x and t x we shall have, 
for the increase of entropy above that of liquid at freezing- 
point, 

The change of entropy in passing from one state to 
another is 

XT X T 

4> - 0, = -j. + 8 - -^ - o, . . . . (144) 

When the condition of the mixture of a liquid and its 
vapor are given by the pressure and value of x, then a table 
giving the properties at even pressures may be conveniently 
used for this work. 

Adiabatic Equation for a Liquid and its Vapor. — Dur- 
ing an adiabatic change the entropy is constant, so that equa- 
tion (144) gives 

^ 1 + e t = ^. + e, (145) 

When the initial state, determined by x x and ^ or p lt is 
known and the final temperature £,, or the final pressure p % , 
the final value x^ may be found by equation (145). The 
initial and final volumes may be calculated by the equations 

v x = x 1 u l -\- a and z\ = xji 2 -f- a. . . (146) 



Il8 THERMODYNAMICS OF THE STEAM-ENGINE. 

Tables of the properties of saturated vapors commonly 
give the specific volume s, but 

S = U -f- (J. 

Values of cr will be found on page ioi. 

Problems in which the initial condition and the final tem- 
perature or pressure are given may be solved directly by aid 
of the preceding equations. Those giving the final volume 
instead of the temperature or pressure can be solved only by 
approximations. An equation to an adiabatic curve in terms 
of/ and v cannot be given, but such a curve for any particular 
case may be constructed point by point. 

Clausius and Rankine independently and at about the 
same time deduced equations identical with equations (144) 
and (145), but by methods each of which differed from that 
given here. 

In the discussion of the specific heat h of a saturated 
vapor, it appeared that the expansion of dry saturated steam 
in a non-conducting cylinder would be accompanied by partial 
condensation. The same fact may be brought out more 
clearly at this place. 

For example, one pound of dry steam at 100 pounds abso- 
lute pressure will have the values 

. t x = 327°-58 F., ^ = 884.0, 0,= 0.4733, x x — 1. 

If the final pressure is 15 pounds absolute, we have 

t % = 213°. 03 F., r, = 965.1, 6^ = 0.3143; 

whence 

884.0 965.1.*' , 

^ + 0.4733 = 6^+0.3143; 

.-. x 9 = 0.8948. 

On the other hand, h is positive for ether, and partial 
condensation takes place during compression in a non-con- 
ducting cylinder. 



20° C 


• > 


r a = 72 


.26, 2 = 


= 0. 


2045; 


93.12 
283.7 


+ 


0.0191 - 


72.26.ar, 

393-7 

= 0.724. 


+ 


0.2045, 



SA TURA TED VAI ORS. 1 1 9 

For example, let the initial condition be 
t x = io° C, r l = 93.12, 0, = 0.0191, x x = I, 
and let the final conditions be 

then 



and 



Equation (145) applies to all possible mixtures of a liquid 
and its vapor, including the case of x x = o or the case of liquid 
without vapor, but at the pressure corresponding to the tem- 
perature according to the law of saturated vapor. When 
applied to hot water, this equation shows that an expansion 
in a non-conducting cylinder is accompanied by a partial 
vaporization. 

There is some initial state of the mixture such that the 
value of x shall be the same at the beginning and at the end, 
though it may vary at intermediate states. To find that value 
make x^ = x x in equation (145) and solve for x xf which gives 

6—6 

*i = 7 — f (147) 

T ~ T 

The value of x x for steam to fulfil the conditions given varies 
with the initial and final temperatures chosen, but in any case it 
will not be much different from one half. It may therefore 
be generally stated that a mixture of steam and water, when 
expanded in a non-conducting cylinder, will show partial con- 
densation if more than half is steam, and partial evaporation 
if more than half water. If the mixture is nearly half water 
and half steam, the change must be investigated to determine 
whether evaporation or condensation will occur; but in any 
case the action will be small. 



120 THERMODYNAMICS OF THE STEAM-ENGINE. 

External Work during Adiabatic Expansion. — Since 
no heat is transmitted during an adiabatic expansion, all of 
the intrinsic energy lost is changed into external work, so 
that, by equation (134), 

W = E x - E, = -gfa - q, + x x p x - x,p 2 ). . (148) 

For example, the external work of one pound of dry steam 
in expanding adiabatically from 100 pounds to 15 pounds 
absolute is 

W= 778(297.9 — 181. 8 -f 1 X 802.8 — 0.8948 X 892.6) 
— 935 1 5 foot-pounds. 

The adiabatic curve cannot be well represented by an 
exponential equation; for if an exponent be determined for 
such a curve passing through points representing the initial 
and final states, it will be found that the exponent will vary 
widely with different ranges of pressure, and still more with 
different initial values of x\ and that, further, the inter- 
mediate points will not be well represented by such an 
exponential curve even though it passes through the initial 
and final points. 

This fact was first pointed out by Zeuner, who found that 
the most important element in determining n was x x , the 
initial condition of the mixture. He gives the following 
empirical formula for determining n, which gives a fair 
approximation for ordinary ranges of temperature: 

»= 1.035 +0.100^ (149) 

There does not appear to be any good reason for using an 
exponential equation in this connection, for all problems can 
be solved accurately by the method given, and the action of 
a lagged steam-engine cylinder is far from being adiabatic. 
An adiabatic line drawn on an indicator-diagram is instructive, 
since it shows to the eye the difference between the expan- 
sion in an actual engine and that of an ideal non-conducting 



SA T URA TED VAPORS. 1 2 1 

cylinder; but it can be intelligently drawn only after an 
elaborate engine test. For general purposes the hyperbola is 
the best curve for comparison with the expansion curve of an 
indicator-diagram, for the reason that it is the conventional 
curve, and is near enough to the curve of the diagrams from 
good engines to allow a practical engineer to guess at the 
probable behavior of an engine, from the diagram alone. It 
cannot in any sense be considered as the theoretical curve. 

EXAMPLES. 

1. Calculate the pressure, heat of the liquid, total heat, 
heat of vaporization, specific volume, etc., at several tempera- 
tures for the vapors for which the data and equations are 
given, and compare with results given in the Tables of the 
Properties of Saturated Steam. 

2. Find the external work of expansion of a fluid, follow- 
ing the law given by the equation pv*, which has the initial 
volume 3 cubic metres and the initial pressure 4 atmos- 
pheres, and which expands till the pressure becomes one 
atmosphere. Ans. 133660 kgm. 

3. A pound of steam and water at 150 pounds pressure is 
0.6 steam; what is the increase of entropy above that of water 
at 32 F. ? Ans. 1.1442. 

4. A kilogram of chloroform at ioo° C. is 0.8 vapor; what 
is the increase of entropy above that of the liquid at 0° C. ? 

Ans. 0.1959. 

5. The initial condition of a mixture of water and steam 
is / = 320 F., ^ = 0.8; what is the final condition after 
adiabatic expansion to 212 F. ? Ans. 0.7398. 

6. The initial condition of a mixture of steam and water 
is p = 3000 mm., x = 0.9; find the condition after an adia- 
batic expansion to 600 mm. Ans. 0.8278. 

7. A cubic foot of a mixture of water and steam, x = 0.8, 
is under the pressure of 60 pounds by the gauge. Find its 
volume after it expands adiabatically till the pressure is 



122 THERMODYNAMICS OF THE STEAM-ENGINE. 

reduced to io pounds by the gauge; also the external work 
of expansion. Ans. 2.6857 cu - ft. and 9981 ft. -lbs. 

8. Three pounds of a mixture of steam and water at 120 
lbs. absolute pressure occupy 4.5 cu. ft. How much heat 
must be added to double the volume at the same pressure 
and what is the change of intrinsic energy ? 

Ans. 1065 B.T.U. ; 750400 ft. -lbs. 

9. A test of an engine with the cut-off at 0.106 of the 
stroke, and the release at 0.98 of the stroke, and with 4.5 per 
cent clearance, gave for the pressure at cut-off 62.2 pounds 
by the indicator, and at release 6.2 pounds; the mixture in 
the cylinder at cut-off was 0.465 steam, and at release 0.921 
steam. Find (1) condition of the mixture in the cylinder at 
release on the assumption of adiabatic expansion to release; 
(2) condition of mixture on the assumption of hyperbolic 
expansion, or that/z; =/ 1 ^ 1 ; (3) the exponent of an exponen- 
tial curve passing through points of cut-off and release; (4) 
exponent of a curve passing through the initial and final 
points on the assumption of adiabatic expansion; (5) the 
piston displacement was 0.7 cubic feet, find the external work 
under exponential curve passing through the points of cut-off 
and release; also under the adiabatic curve. 

Ans. (1) 0.472; (2) 0.524; (3) n = 0.6802 ; (4) n — 1.0589; 
(5) 3093 and 2120 ft.-lbs. 



CHAPTER VII. 
SUPERHEATED VAPORS. 

A DRY and saturated vapor, not in contact with the liquid 
from which it is formed, may be heated to a temperature 
greater than that corresponding to the given pressure for the 
same vapor when saturated; such a vapor is said to be super- 
heated. When far removed from the temperature of satura- 
tion such a vapor follows the laws of perfect gases very 
nearly, but near the temperature of saturation the departure 
from those laws is too great to allow of calculations by them 
for engineering purposes. 

In the case of superheated steam various provisional 
characteristic equations have been proposed for use until the 
necessary experimental investigation shall give the data for a 
true theory. The theory given here was proposed by Zeuner. 
It is convenient for calculation and appears to give good 
results. 

Substituting in the characteristic equation for a gas 

pv = RT, 
the value of R from equation (64) gives 

Cp— C v C p K — I 

^ = __ r = _ ___ r . . . (I50) 

The form of characteristic equation proposed by Zeuner 
for superheated steam is 

pv = a k ~lT T - Cp "- • ' ••• (I5I) 

123 



124 



THERMODYNAMICS OF THE STEAM-ENGINE. 



The specific heat at constant pressure c P is assumed to be 
constant, k is a constant suggested by the ratio k of the 
specific heats of a gas; but it will be shown that the specific 
heat at constant volume, determined from the equation (151), 
is a variable; consequently k cannot be the ratio of the specific 
heats of superheated steam. C and a are constants that are 
to be determined from the known properties of saturated and 
superheated steam. 

Partial Differential Coefficients. — From the character- 
istic equation (151) may be deduced the partial differential 
coefficients 



(dt\ Apk 



dt\ 



Ik - 1) 

Avk , ap a ~ l AkC 

+ -T7T ;v • • ■ ( J 53) 



dpj v c p (k — 1) c p {k — I)' 



Application of the First Law. — The first law of ther- 
modynamics may be conveniently applied by using equation 
(48), substituting for n and their values, in terms of the 
specific heats, from the proper equations on page 13; thus: 



n = c 



do\ 



A ( \dp) v \dv)^ 



dt 



d 



v ^pi: 

idt\ ~ 



dp 



ldt\ 
0=C kdv)j 



d 



c<-\ 



dv 



= A. . . (154) 



In applying this equation it is convenient to substitute for 



7. \ 

-r) from equation (152), perform the differentiation indi- 

dvi p 

cated, and then simplify the result, giving 



SUPERHEATED VAPORS. 



125 



Ak 
k- 1 



(dt 
d V c M> v A 
dv 



= A; . . . (155) 



'[*S)j 



dv 



t = (I56) 



If this expression is integrated we get a new expression 
for the partial differential coefficient \-j -J instead of the 
somewhat complicated expression of equation (152), namely, 



Idt 

\db 



\ _ 



Av 



(157) 



\dp) v c v {k— 1)' 
Application of the Second Law. — Equation (55), 



deduced by the successive application of the two laws of 
thermodynamics, can be most conveniently used in this place. 
Substituting the values of the partial differential coefficients 
from equations (152) and (157) gives 



C-, — Cj,C 



{k - i) 2 T 



P^v 



?=!+< 



k 
(k 



Apv ' 
i) 2 T 



Apv' ' 



(158) 



which gives the method of calculating the specific heat at 
constant volume when c p and k are known. 

Exponent a. — Equating the values of the differential 
coefficient given by equations (152) and (157), we have 



Av 



Avk 



+ 



ap a ~'AkC 



c v {k — 1) c p {k — 1) c p (k — 1) * 



126 THERMODYNAMICS OF THE STEAM-ENGINE. 

Simplifying and solving for a 7 we have 

a = gJ k _ Sf). 

kCp a ^ cj 

c. 
Substituting the values of — and of Cp a from equations (158) 

and (151), 

, {k - iy t 

k — I — Cj, : -; 

pv y k Apv 

a = ~k k- \T ; 

k- 1 
•*• az=z —k~ 059) 

Characteristic Equation for Superheated Vapor. — Sub- 
stituting the value deduced for a in equation (151) gives, for 
the characteristic equation for superheated steam, 



f v = A^ir T ~ Cft (l6o) 



Thermal Capacities. — From equations (n), (152), and 
(158). 

I-J'J- ,)(*) _. (^-i)' T Apk 
\, X )\dvl t ~ ■"* k Apvc t (k-\)' 

.-. l = clk-iy- (161) 



From equations (15), (157), and (158), 



c p \[dt\ (k- i) a T Av 



t - C - u \c v " 1 )\dp)- CvCp k Apvc v {k-i)' 

k-iT 



a • 



-m = c,— r j (162) 



S UPE RHEA TED VA PORS. 1 2 7 

From equations (17) and (152), 



fdt \ Apk 



From equations (18) and (157), 



ldt\ Av 

n = c "\dp) v = k- 7 ( l6 4) 



General Equations. — Substituting the values of /, m, n, 
and in equations (5), (6), and (7) gives, for the general equa- 
tions for superheated steam, 

dQ = c v \dt+{k- i)—dv} ; . . (165) 

dQ= c p \dt - — =^- -^/} ; • • (166) 

dQ = y--j{vdp + 6pdv\ (167) 



It is instructive to compare these equations with the 
general equations (70), (71), and (72) for perfect gases, which 
may be written, 

dQ= c v \dt-\-( K — 1)— dv\ ; . . (168) 

C k — i T ) 
dQ= Cp\dt — ~zdp\^ \ . . (169) 

dQ = ^——^{vdp -{- K p dv\ (170) 



128 THERMODYNAMICS OF THE STEAM-ENGINE. 

To obtain equation (170), equation (72) may be written,, 



v p 

dQ = c v -^dp + c P gdv ; 



It is to be remarked that equation (165) is not useful irt 
its present form, since c v is a variable, but it is written for 
symmetry in comparison with equations (168), (169), and 

07O). 

Entropy. — Equation (166) gives 



dQ j dt k-idp , 

rf0 = T= ,,j T _.__J ; . . . . ( I7I) 

( , T k- 1 , p ) 



which is to be compared with equation (89), page 71, for gases. 
Equations (165) and (167) cannot conveniently be used for 
calculating change of entropy. 

Value of h. — The characteristic equation for superheated 
vapor is intended to apply to all degrees of superheating, 
approaching at one limit the condition of a gas, and at the 
other that of saturated vapor. For a mixture of a liquid and 
its vapor we have, from equation (144), 

defy = d(^) +d0= d{^) + 4^dt, 

or for saturated steam with x = 1 

dcf> = -rpdt -\- d[ y) = -f\cdt -\-dr — -^dtj. . (173) 



SUPERHEATED VAPORS. 1 29 

Equations (171) and (173) should both be true for dry 
saturated vapor, whence 



k— 1 T dp\ dr 



c '\ l --T~JdI)= c + dt-T- ■ ■ < I74 > 



By equation (129) the right-hand member of equation 
(174) is equal to k, the specific heat of saturated vapor; con- 
sequently 

k — 1 c* — h f 

(175) 



~p~di cp 



Superheated Steam. — Regnault gives as the results of 
three experiments on the specific heat of superheated steam 
at constant pressure 

0.481 1 1, 0.48080, 0.47963, 

and for the mean value 

c p = 0.4805. 

With this value of cp and the known values of the other 
factors, determined from the properties of saturated steam, 
the following values of k were calculated: 

Pressure, pounds ) 

. . \ 5 50 100 200 300 

on the sq. in. ) ° 

k 1.33 1.332 1.330 1.324 1. 316 

Zeuner assumed for the constant k the value 

k= i = !-333 +, 



-130 THERMODYNAMICS OF THE STEAM-ENGINE. 

which may be compared with the ratio of the specific heats of 
air, 

*•= 1.405. 

With this assumed value of k and the known values of A 
and c p the coefficient of T in the characteristic equation (160) 
becomes: 

c p k— I 



French system, -j — -7 — =B= 51.28; 

r fo T 

English system, -j , = B = 93.46. 



The specific volume of saturated steam under atmospheric 
pressure and at boiling-point is 26.60 cubic feet or 1.661 
cubic metres. Solving equation (160) for C, 



'T-pv 



A k 
C = 



k - 1 
p k 



and therefore we have: 

French system, 

51.28 X 373-7 — 10333 X 1. 661 

L = ' i = 190.4; 

I0333 4 

English system, 

^ 93-46" X 672.7 — 2116.32 X 26.60 

C = t = 971. 

2116.32 1 

Substituting the constants in the characteristic equation 
gives: 

French system, pv = $i.sT— igSp* (176) 

English system, pv = 93.5 T — 97 ip* ( l 77) 



S UPERHEA TED VA PORS. 1 3 1 

Zeuner's constants for equation (176) differ from those 
given, since he used 424 for the mechanical equivalent of one 
calorie, and 273 for the absolute temperature of freezing- 
point. 

In using these equations for superheated steam it is to be 
remembered that the pressures are specific pressures — i.e., 
kilograms per square metre or pounds per square foot — 
whereas the pressures of saturated steam are commonly stated 
in millimetres of mercury or in pounds on the square inch. 

The form of the equation lends itself to the ready calcula- 
tion of volume or temperature; but the calculation of pressure 
can be made only by successive approximations. 

For example, the specific volume of steam having the 
pressure of 100 pounds by the gauge and a temperature of 
400° F. is 

_ 93.5 T- 971/)* 93.5 X 860.7 — 971(144 X 1 14-7)* . 
p 144 X 1 14-7 

. • . v = 4.20 cubic feet. 

For example, the pressure of superheated steam having a 
temperature of 400 F. and a specific volume of 5 cubic feet 
is approximately 

93.5 T 93.5 X 860.7 ,,. . 
p = — = IOIOO 

r v 5 

more accurately it is 

93.5 X 860.7 - 971 X 16095* 
/= = 13900; 

and a third approximation is 



93 5 X 860.7 — 971 X 13900* 
p = — = 13990, 

p = 97.1 pounds per square inch absolute. 



132 THERMODYNAMICS OF THE STEAM-ENGINE. 

Specific Meat at Constant Volume. — The specific heat 
of superheated steam at constant volume may be calculated 
by applying equation (158) to the case of saturated steam. 
The following table gives the values obtained at several 
pressures: 

SPECIFIC HEAT OF SUPERHEATED STEAM. 

Pressures, pounds ) 

r . . \ 5 50 100 200 300 

per square inch, ) • 

Specific heat, c v9 O.351 O.348 0.346 O.344 0.341 

This table develops the fact already mentioned that the 
specific heat of superheated steam at constant volume, 
deduced from the form of the characteristic equation (160) 
and the known properties of saturated and superheated steam, 
is a variable. This conclusion applies properly to steam 
that is only slightly superheated, whereas our experimental 
knowledge of the properties of superheated steam relates to 
steam that is superheated to a marked degree. 

Intrinsic Energy. — The combination of the equation 

dQ = A(dE+pdv) 
with equation (167) gives 

dE — y—^vdp+pdv) = J"— ^0^) 5 

••• B-B^/^-/^,. . . ( I7 8) 

an equation identical in form with that for a perfect gas. 

It is convenient to calculate the intrinsic energy from 
the freezing-point of water, using a combination of equation 
(178) and equation (134), 



S UPERHEA TED VA PORS 133 

for saturated vapor. The increase of intrinsic energy due to 
heating a liquid from freezing-point to the temperature t and 
entirely vaporizing it is 

The increase of energy due to superheating the vapor under 
the constant pressure/ so that the specific volume increases 
from that for saturated steam to that for superheated steam is 

pv ps p(v — s) 



The total increase of energy is 

£ =^F + j(? + ^ • • • (i79) 

Total Heat of Superheated Vapor. — By the total heat 
of superheated vapor is meant the heat required to change 
one unit of weight of the liquid at freezing-point into super- 
heated vapor having a given temperature. It may be sep- 
arated into three parts: the heat of the liquid q, the heat of 
vaporization r> and the heat required to superheat the steam, 

c £t* — 0» 

in which t s is the temperature of the superheated steam and t 
is the temperature of saturated steam at the same pressure. 
The total heat is consequently 

9+r+cJt M -1) (i 80) 

Comparison with Experiments. — Experiments on the 
specific volume of superheated steam were made by Hirn,* 

* Thdorie M^canique de la Chalenr. 



134 



THERMODYNAMICS OF THE STEAM-ENGINE, 



from the report of which Zeuner selected the experimental 
data in the following table. The specific volume has been 
calculated by aid of equation (176), and placed in the table 
opposite the experimental results to show the comparison of 
the characteristic equation with experiments. 



SPECIFIC VOLUME OF SUPERHEATED STEAM. 



Pressure 

in 

atmospheres. 


Temperature. 
Centigrade. 


Specific Volume. 
Cubic meters. 


Hirn's 
experiments. 


Equation (176). 


I 
I 

3 
4 
4 

4 

5 
5 


Il8. 5 

141 

200 

165 

200 

246 

162.5 

205 


1.74 

1. 85 

0.697 

0.4822 

0.522 

o.5752 

o.3758 

0.414 


1-75 

I.87 

O.699 

O.476 

O.520 

0.577 
O.376 
O.418 



The table on the next page shows that the characteristic 
equation for superheated steam applies fairly well to the limit- 
ing case of saturated steam. The values in column 2 were 
taken directly from the table of the properties of saturated 
steam, and the corresponding quantities in column 3 were cal- 
culated by aid of equation (177). The entropies in column 4. 
are calculated by the expression 



r 



Column 5 is obtained by calculating by equation (172) the 
change of entropy from freezing-point to the given pressure 
and corresponding temperature, and adding it to the entropy 
at freezing-point; the change of entropy is negative and when 
added gives a decreasing value to the entropy as the pressure 
rises. 



S UPERHEA TED VA P OPS. 
APPLICATION TO SATURATED STEAM. 



135 





Specific 


Volumes, 


Enti 




Absolute 


Cubic Feet. 


opy. 


Pressure, 
Pounds per 


















Square Inch. 


Tabular 


Equation 


Equation 


Equation 




Value. 


( J 77)- 


(143). 


(172). 


1 


2 


3 


4 


5 


14.7 


26.60 


29.6 


1.7484 


1-752 


30 


13-59 


13-7 


I. 689I 


1.704 


60 


7.O96 


7.12 


I.634O 


1.641 


IOO 


4-403 


4-38 


1-5945 


1.598 


ISO 


3. on 


3.00 


1 . 5649 


1.568 


200 


2.294 


2.30 


1.5446 


1.546 


300 


1-554 


i-57 


1.5262 


1.517 



Adiabatic Line. — During an adiabatic change the entropy 
remains constant; consequently from the general equation 
(165) we have 



o = 




dt 1 iu \ C dv 

Y + c v (k- 1) / -; 



{k - 1) log,- ; 



Tv k ~ x = T x v*-\ 



(181) 



which is deduced in the same way as the corresponding equa- 
tion for a perfect gas, but differs in that k is an arbitrary con- 
stant, while the equation for perfect gases has in the exponent 
k the ratio of the specific heats. 

From equation (166) we may deduce in a similar manner 
the equation 



Tp = = t,a 



(182) 



Equation (167) is not in convenient form for treatment in 



136 THERMODYNAMICS OF THE STEAM-ENGINE. 

the manner used for deducing the two preceding equations, 
but from those equations we may readily deduce 

^v k =p i v 1 k , (183) 

which may be considered to be the typical adiabatic equation. 
The external work during an adiabatic expansion may be 
calculated by an equation having the same form as that 
deduced for perfect gases, i.e., 

-=&{.- tin.. . . m 

provided that the vapor remains superheated to the end of 
the expansion. 

If a vapor is not very strongly superheated it is liable to 
become saturated and moist during an adiabatic expansion, 
and in that case an extension of the method used for a mix- 
ture of a liquid and its vapor must be employed. The 
increase of entropy of superheated vapor above that of the 
liquid at freezing-point may be divided into three parts: 1st, 
that due to heating the liquid; 2d, that due to vaporizing 
the liquid; and 3d, that due to superheating the steam. The 
first two parts have already been discussed in the chapter on 
saturated vapors; the third part is represented by 



Cdt T s 



where c is the specific heat of the vapor at constant pressure 
and T s is the temperature of the superheated vapor, while T x 
is the temperature of saturated vapor at the given pressure. 
Assuming that the steam is moist at the final pressure/,, and 
temperature T„ we may calculate the condition x % by aid of 
the equation 

T T X T 

-^ + 6 x + Cp ^g, ~ = ~J^ + 6 r ' ' ( l8 5) 



SUPERHEATED VAPORS, 137 

The specific heat of superheated steam is 

c p — 0.4805. 

For example, let the initial pressure be 100 pounds abso- 
lute per square inch and the initial temperature be 400 F. ; 
required the condition of the steam after an adiabatic expan- 
sion to 15 pounds absolute. Here we have 

t x = 327°-6, r, = 884.0, e i = 0.4733, 

* a =2i3°.o, r a = 965.1, #. = 0.3143; 

884.0 , ,01 86o -7 9 6 5- I * 3 ■ 

. ' • ^ + °' 4733 + a48 ° 5 l ° g ° ^87 3 = ^7 + °'3 I 43 ! 

. • . ^ = 0.923. 

The external work for such an adiabatic expansion is 
obtained by taking the difference of the initial and final 
intrinsic energies, which may be calculated individually by 
equations (179) and (134). 

For example, with the conditions of the preceding problem 
we have 



■ 93-5 7\- 97^1* ._ 93-5 X 860.7-971 x 14400 
Vl " A 14400 

= 4.85 cubic feet. 

Consequently the intrinsic energy is 



= '4400(4-85 -4-403) + 778(297 . 9 + 8o2 . 8) = 87570O ft ..i bs , 



"3" 



I 



138 THERMODYNAMICS OF THE STEAM-ENGINE. 

The final intrinsic energy is . 

E* = 77K<1* + *,P>) = 778(i8i.8 + 0.923 x 892.6) 

= 782400 foot-pounds. 

so that the external work is 

E 1 — E^ = 875700 — 782400 = 93300 foot-pounds. 

Isoenergic Line. — The equation to this line is obtained 
from equation (178) by making E equal to E l9 so that 

pv=p 1 v 1 , (186) 

which, like the isoenergic line for a perfect gas, is the equa- 
tion to a rectangular hyperbola. 

The external work during an isoenergic expansion is 

W= fpdv = pp log,^ = p x v y log.£. . (187) 

Since all the heat applied is expended in external work, 

Q = AW. (188) 

Isothermal Line. — The equation to the isothermal line 
for a superheated vapor is obtained by making T a constant 
in the characteristic equation 

f k — t k ~ z 

* V = A— T - C > h > 



so that 

pv=p,v x - C\p k -A * |. . . . (189) 



= A*i- C\p k -A * ). ... (1; 



SUPERHEATED VAPORS. 1 39 

The heat applied during an isothermal change is obtained 
by integrating equation (166) with T constant, giving 

e = ^i J ^ riog <A (I90) 

But we have in general 

Q = A(E,-E 1 +W), 

so that the external work is 



W=%+E l -E t , 



which may be reduced by equations (190) and (178) to 

w k A g 'A k-\ k-i ' ^ 9I > 



Properties of Sulphur Dioxide. — One of the most inter- 
esting and important applications of the theory of superheated 
vapors is found in the approximate calculation of properties 
of certain volatile liquids which are used in refrigerating- 
machines, and for which we have not sufficient experimental 
data to construct tables in the manner explained in the 
chapter on saturated vapors. 

For example, Regnault made experiments on the pressures 
of saturated sulphur dioxide, ammonia, and carbon dioxide, 
but did not determine the heat of the liquid nor the total 
heat. He did, however, determine some of the properties of 
these substances in the gaseous or superheated condition, 
from which it is possible to construct the characteristic equa- 
tions for the superheated vapors. These equations can then 
be used to make approximate calculations of the saturated 
vapors, for such equations are assumed to be applicable down 



140 THERMODYNAMICS OF THE STEAM-ENGINE. 

to the saturated condition. Of course such calculations are 
subject to a considerable unknown error, since the experi- 
mental data are barely sufficient to establish the equations for 
the superheated vapors. 

The specific heat of gaseous sulphur dioxide is given by 
Regnault * as 0.15438, and the coefficient of dilatation as 
0.0039028. The theoretical specific gravity compared with 
air, calculated from the chemical composition, is given by 
Landolt and Bornstein f as 2.21295. Gmelin \ gives the 
following experimental determinations: by Thomson, 2.222 ; 
by Berzelius, 2.247. The figure 2.23 will be assumed in this 
work, which gives for the specific volume at freezing-point 
and at atmospheric pressure 



0.7735327 _„ ,. 

v = ■ = 0.347 cubic metres. 

2.23 ° ' 



The corresponding pressure and temperature are 10333 an d 
273°.7 C. 

Now the coefficient of dilatation is the ratio of the increase 
of volume at constant pressure, for one degree increase of 
temperature, to the original volume. Writing the equation 
(160) in the following form, 



P v = C j [ aT- Cp«, (192) 



and applying it at o° C. and i° C, we have 



r 



Si 
A 



* M/moires de V Institut de France, tome xxi, xxvi. 
f Physikalische-chemische Tabellen. 
\ Watt's translation, p. 280. 



SUPERHEATED VAPORS. 141 



a 



v a pj>; 

Substituting the known values and solving for a, we 
obtain 0.212; but the equation obtained from the equation 
(192) with this figure does not agree well with Regnault's 
experiments on the compressibility of sulphur dioxide. If, 
instead, we make 

a = 0.22, 

then by equation (192) the coefficient of dilatation becomes 
0.00404, and it will be shown later that the equation deduced 
with this value agrees quite well with the experiments on 
compressibility. 

The coefficient of T in equation (192) is therefore 

0.15438 X 426.9 X 0.22 = 14.5, 

and the coefficient of p a is 

14.5 X 273.7 - 10333 X O.347 



0.22 
10333 



= 48 nearly ; 



so that the equation becomes 

pv= 14.5^- 48/ 022 (193) 

Regnault found for the pressures 

/>, = 697.83 mm. of mercury, 
A = 1341.58 " " " 

and at y°.y C. the ratio 

—^ = 1.02088. 



142 THERMODYNAMICS 01 THE STEAM-ENGINE. 

Reducing the given pressures to kilograms on the square 
metre, and the temperature to the absolute scale, and applying 
to equation (192), we obtain 1.016 instead of the experi- 
mental value for the above ratio. 

Regnault gives for the pressure of saturated sulphur diox- 
ide, in mm. of mercury, the equation 

log/ = a — ba n — cfi n ; 

a = 5.6663790 ; 
log b = 0.4792425 ; 
\ogc = 9.1659562 — 10; 
log a = 9.9972989 — 10 ; 
log/? — 9.98729002 — 10; 

n = t + 28 C. 

Applying equation (no), page 92, to this case, 

log a = 9.9972989 ; 
log/? = 9.98729002 ; 
\ogA — 8.6352146; 
log^= 7.9945332; 
n = t + 28 C. 

The specific volume of saturated sulphur dioxide may be 
calculated by inserting in equation (193) for the superheated 
vapor the pressures calculated by aid of the above equation. 
The results at several temperatures are a-s follows: 

/ — 30 C. o + 30 C. 

s 0.8292 0.2256 0.0825 

Andreeff* gives for the specific gravity of fluid sulphur 

* Ann. Chem. Pharm., 1859. 



SUPERHEATED VAPORS. 1 43 

dioxide 1.4336; consequently the specific volume of the 
liquid is ■ 

<r = 0.0007. 

The value of r, the heat of vaporization, may now be cal- 
culated at the given temperatures by equation (128), page 104, 

r = AuT^-, 
at 

in which u = s — cr. 

The results are 

t — 30 C. o + 30 C. 

r 106.9 97.60 90.54 

Within the limits of error of our method of calculation, 
the value of r may be found by the equation 

r = 98 — 0.27/ (194) 

To find the specific heat of the liquid we may use equa- 
tion (174), page 129, 

/ T dp\ dr r 

W— ,■*)-' + *— T- 

At o° C. the specific heat is approximately 

c = 0.4. 

In English units we have for superheated sulphur dioxide 

pv = 26.4T - 1S4P™ 2 , (195) 



144 THERMODYNAMICS OF THE STEAM-ENGINE. 

the pressures being in pounds on the square foot, the volumes 
in cubic feet, and the temperatures in Fahrenheit degrees 
absolute. 

For pressures in pounds on the square inch at tempera- 
tures on the Fahrenheit scale, 

log p — a — ba n — cft n ; 

*= 3-9527847; 
log b = 0.4792425 ; 

log c = 9.1659562 — 10; 

log a = 9.9984994 — 10 ; 

log fi = 9.99293890 — 10; 

n= t+ i8°4 F. 

For the heat of vaporization 

r = I7 6 — 0.27(7 - 32), .... (196) 

and for the specific heat of the liquid 

c = 0.4. 

In applying these equations to the calculation of a table 
of the properties of saturated sulphur dioxide the pressures 
corresponding to the temperatures are calculated as usual. 
Then the heat of the liquid is calculated by aid of the con- 
stant specific heat. The heat of vaporization is calculated by 
aid of equation (196). Next the specific volume is calculated 
by inserting the given temperature and the corresponding 
pressure for the saturated vapor in the characteristic equation 
(193) or (195). Having the specific volume of the vapor and 
that of the liquid, the heat equivalent (Apu) of the external 
work is readily found. Finally, the entropy of the liquid is 
calculated by the equation 

T 
6 = c\og e jr (197) 



SUPERHEATED VAPORS. 145 

Properties of Ammonia. — The specific heat of gaseous 
ammonia, determined by Regnault, is 0.50836. The the- 
oretical specific gravity compared with air, calculated from 
the chemical composition, is given by Landolt and Bornstein 
as 0.58890. Gmelin gives the following experimental deter- 
minations: by Thomson, 0.5931 ; by Biot and Arago, 0.5967. 
For this work the figure 0.597 will be assumed, which gives 
for the specific volume at freezing-point and at atmospheric 
pressure 

0.7735327 

v a = =1.30 cubic metres. 

0.597 

The coefficient of dilatation has not been determined, and 
consequently cannot be used to determine the value of a in 
equation (192). It, however, appears that very consistent 
results are obtained if a is assumed to be \, as for super- 
heated steam. The coefficient of T then becomes 

0.50836 X 426.9 X i= 54-3> 
and the coefficient of/* is 

54.3 X 273.7- 10333 x 1.30 
=1 = 142 ; 

10333* 

so that the equation becomes 

pv= 54.3 T— 142/1 (198) 

The coefficient of dilatation, calculated by the same 
process as was used in determining a for sulphur dioxide, is 
0.00404, which may be compared with that for sulphur 
dioxide. 

Regnault found for the pressures 

p i = 703.50 mm. of mercury, 
A = 1435 3 " " 



14^ THERMO D YNAMICS OF THE STEAM-ENGINE. 

and at 8°.i C. the ratio 



PJ>* 



= 1.0188, 



while equation (198) gives under the same conditions 1.0200. 
For saturated ammonia Regnault gives the equation 

log / = a — ba n — c/3 H ; 

a == 11.5043330; 
log b = 0.8721769; 
log c= 9.9777087—10; 
log a — 9.9996014 — 10 ; 
log fl = 9.9939729 — 10 ; 

n = t + 22 C. ; 

by aid of which the pressures in mm. of mercury may be cal- 
culated for temperatures on the centigrade scale. The 
differential coefficient may be calculated by aid of the equa- 
tion 

log A = 8.1635170— 10; 
logj5 = 8.4822485 — 10; 
log a = 9.9996014 — 10; 
log = 9.9939729 - 10 ; 

« = /+22°C. 

The specific volume of saturated ammonia calculated by 
equation (198) at several temperatures are 

/ - 30 C. o + 30 C. 

s 0.9982 0.2961 0.1 167 

Andreeff gives for the specific gravity of liquid ammonia 
at o° C. 0.6364, so that the specific volume of the liquid is 

a = 0.0016. 



SUPERHEATED VAPORS. 1 47 

The values of r at the several given temperatures, calcu- 
lated by equation (128), are 

t - 30 C. o + 30 C. 

r 325.7 300.15 277.5 

which may be represented by the equation 

r — 300 — 0.8/ ( x 99) 

The specific heat of the liquid, calculated by aid of equa- 
tion (129), is 

c = 1.1. 

In English units the properties o superheated or gaseous 
ammonia may be represented by the equation 

pv — 99T— 540/*, (200) 

in which the pressures are taken in pounds on the square foot 
and volumes in cubic feet, while T represents the absolute 
temperature in Fahrenheit degrees. 

The pressure in pounds on the square inch may be calcu- 
lated by the equation 

log p = a — ba n — c/3 n ; 

a = 9.7907380 ; 

log b = 0.8721769 — 10 ; 

log c = 9-9777087 — 10 ; 

log a = 9.9997786 — 10 ; 

log fi = 9.99665 16 — 10 ; 

n = t-\-7°.6 F. 

The heat of vaporization may be calculated by the equa- 
tion 

r = 540 — o.8(V — 32), ( 2QI ) 

and the specific heat of the liquid is 

c = 1.1. 



*48 THERMODYNAMICS OF THE STEAM-ENGINE. 



EXAMPLES. 

1. What is the weight of one cubic foot of superheated 
steam at 500 F. and at 60 pounds pressure absolute ? 

Ans. 0.107 lb. 

2. Superheated steam at 50 pounds absolute has half the 
density of saturated steam at the same pressure. What is the 
temperature ? Ans. 930°.8 F. 

3. Find the increase of intrinsic energy and increase of 
entropy of a pound of superheated steam, at 100 pounds 
absolute and at 400 F., above the values at 32 ° for water. 

Ans. 876400 ft. -lbs. and 1.6369. 

4. Find the external work of one kilogram of steam in 
expanding adiabatically from the pressure of 3000 mm. of 
mercury and the temperature 300 C. to the pressure of 
2000 mm. Find also the final temperature and volume. 

Ans. (1) 7690 m-kgs. ; (2) 244°./ C. ; (3) 0.8842 cu. ft. 

5. In example 4 find the external work for an isothermal 
expansion from the initial condition to the final volume as 
determined in that example. Ans. 8120 m.-kgs. 

6. Let the initial temperature of superheated steam be 
380 F. at the pressure of 150 pounds absolute. Find the 
condition after an adiabatic expansion to 20 pounds abso- 
lute. Determine also the initial and final volumes. 

Ans. (1)0.8953; (2) 3.094 cu. ft.; (3) 17.83. cu. ft. 

7. In example 9, page 122, suppose that the steam at 
cut-off were superheated io° F. above the temperature of 
saturated steam at the given pressure, and solve the example. 

Ans. (1) 0.8865; (2) 79 . 5 superheating; (3) same as 
before; (4) n — 1.137; (5) 1972 and 1950 ft. -lbs. 



CHAPTER VIII. 

FLOW OF FLUIDS. 

One of the mast important problems in thermodynamics 
is to find the amount of a gas or a vapor which will be dis- 
charged through a given orifice in a unit of time. To make 
the statement of the problem more concrete it will be sup- 
posed that the fluid passes from the 
large cylinder A (Fig. 30) into the h 

smaller cylinder B through a well- 
rounded orifice at C. The cylinders 



pi 



r 



will be supposed each to be at the 

same temperature as the fluid in it, so that there will be no 
communication of heat to or from the walls of the cylinders. 
The process is clearly non-reversible, so that only the first 
law of thermodynamics can be used, and, as in equation (43), 

dQ = A{dE + dW + dK), 

a term must be added to represent the kinetic energy due to 

ordinary motion of translation. 

Let it be supposed that there is a frictionless piston in 

each cylinder; the piston in A exerts the pressure p l on the 

fluid in front of it, and the piston in B has on it the fluid 

pressure p^. Each unit of weight of fluid passing from A 

through the orifice has the work p l v l done on it, while each 

pound entering the cylinder B does the work pjv % . The 

assumption of pistons is merely a matter of convenience, and 

if they are suppressed the same conditions with regard to 

external work will hold. 

149 



ISO THERMODYNAMICS OF THE STEAM-ENGINE. 

If the velocity in A is V x the kinetic energy of one unit 

V 3 
of weight in that cylinder is — ; the kinetic energy in B is 

V 

— - for a velocity V„. 

2 g 

The intrinsic energies in A and B are E x and E^. If there 
is no heat communicated to or from the fluid the sum of the 
intrinsic energy, external work, and kinetic energy must 
remain constant, so that 



V' 1 V 

£, + A^ + ^ = £> + pp* + ~ ; • • (202) 



this is the fundamental equation for the flow of a fluid. 

It is proper to include a term for the gain or loss of heat 
at the orifice, which is commonly made of metal and is in 
metallic contact with the walls of the cylinders, and will not 
have the temperature of the fluid in it; but the amount of 
heat that can be communicated to the fluid at the orifice is 
too small to take account of, consequently the term is not put 
into the fundamental equation. 

Usually the velocity in the large cylinder A is small and 
the term depending on it may be neglected. Solving for the 
term depending on the velocity in B and dropping the sub- 
script, we have 

- = £>-£> + A^» - A*V • • • (203) 

Incompressible Fluids. — There is little if any change of 
volume or of intrinsic energy in a liquid in passing through 
an orifice under pressure, so that the equation of flow becomes, 
in this case 

P 

-= (A-A>i (204) 



FLOW OF FLUIDS. 151 

If the difference of pressure is due to a difference of level 
or head, h, we have 

A — A = *r. 

where y is the density, or weight of a unit of volume, and is 
the reciprocal of the specific volume; consequently equation 
(204) reduces to 

— =h, ....... (205) 

which is the usual equation for the flow of a liquid through a 
small orifice. 

Flow of Gases. — The intrinsic energy of a unit of weight 
of a gas is 

E = - pV 



K— V 

so that the equation for the flow of a gas is 
V * P^ P* v * 

-^ = lFzr : i - 7Fzr } i +A", -A*,; 

V K 

•'• — - = ^— iCA^-A*'.) ( 2o6 ) 

For an adiabatic transformation 

AV=M"5 (207) 

• • A* 9 =A*,@* * =P* v lj}~' 
so that equation (206) may be reduced to 



1 — 



2g ri l K — I 



(£)""']. • • • (208) 



152 THERMODYNAMICS OF THE STEAM-ENGINE. 



If the area of the orifice is a y then the volume discharged 
per second is 

and the weight discharged per second is 

aV 



w = 



v n 



where v 9 is the specific volume at the lower pressure / a . 
Substituting for V from equation (208) and for v 7 from (207), 



^a 



w = a 



ZgPy * 



V x K — I 









But from the characteristic equation 



v x = 



so that 







^P 1 



K+l 



A 

PJ J 



(210) 



The equations deduced for the flow of air apply to the 
flow from a large cylinder or reservoir into a small straight 
tube through a rounded orifice. The lower pressure is the 
pressure in the small tube and differs materially from the 
pressure of the space into which the tube may deliver. In 
order that the flow shall not be affected by friction against 
the sides of the tube it should be short — not more than once 



FLOW OF FLUIDS. 153 

or twice its diameter. The flow does not appear to be 
affected by making the tube very short, and the degree of 
rounding is not important; the equations for the flow of both 
air and steam may be applied with a fair degree of approxi- 
mation to orifices in thin plates and to irregular orifices. 

Professor Fliegner* made a large number of experiments 
on the flow of air from a reservoir into the atmosphere, with 
pressures in the reservoir varying from 808 mm. of mercury 
to 3366 mm. He used two different orifices, one 4.085 and 
the other 7.314 mm. in diameter, both well rounded at the 
entrance. 

He found that the pressure in the orifice, taken by means 
of a small side orifice, was 0.5767 of the absolute pressure in 
the reservoir so long as that pressure was more than twice 
the atmospheric pressure; under such conditions the pressure 
in the orifice is independent of the pressure of the atmosphere. 

P 
If the ratio — is replaced by the number 0.5767 and if k is 

replaced by its value 1.405 in equation (210) we shall have 
for the equation for the flow of a gas 

W = O.A$22a\ / -£ -A=r (2Il) 

V R VT X 

Far the flow into the atmosphere from a reservoir having 
a pressure less than twice the atmospheric pressure Fliegner 
found the empirical equation 



VfV-'' - " 



• " . - r, T r " , ■ ■ (212) 

where p a is the pressure of the atmosphere. 

These equations were found to be justified by a compari- 
son with experiments on the flow of air, made by Fliegner 
himself, by Zeuner, and by Weisbach. 

* Der Civilingenieur \ vol. xx, p. 14, 1874. 



154 THERMODYNAMICS OF THE STEAM-ENGINE. 

Although these equations were deduced from experiments 
made on the flow of air into the atmosphere, it is probable 
that they may be used for the flow of air from one reservoir 
into another reservoir having a pressure differing from the 
pressure of the atmosphere. 

Fliegner's Equations for Flow of Air. — Introducing the 
values for g and R in the equations deduced by Fliegner, we 
have the following equations for the French and English 
systems of units: 

French units. 
A 



/, > 2p ai w = o.$9$a 



VT.' 



A < 2/ a , ,= o^oaJ^^JA 



English units. 
p L >2f> a , w = o.$ioa-f^= ; 



p x = pressure in reservoir; 
p a = pressure of atmosphere; 

T x = absolute temperature of air in reservoir (degrees centi- 
grade, French units; degrees Fahrenheit, English 
units). 
In the English system p y and p a are pounds per square 
inch, and a is the area of the orifice in square inches, while 
w is the flow of air through the orifice in pounds per second. 
If desired, the area may be given in square feet and the pres- 
sures in pounds on the square foot, as is the common conven- 
tion in thermodyanmics. 



FLOW OF FLUIDS. 155 

In the French system w is the flow in kilograms per 
second. The pressures may be given in kilograms per square 
metre and the area a in square metres; or the area may be 
given in square decimetres or square centimetres, and the 
pressures in kilograms on the same unit of area used in con- 
nection therewith. If the pressures are in millimetres of 
mercury, multiply by 13.5959; if in atmospheres multiply 
by 10333. 

Theoretical Maxima. — It is interesting to investigate the 
conditions that give the maximum discharge of air as calcu- 
lated by equation (210), neglecting for the moment Fliegner's 
experimental limit of the ratio of pressures. For this purpose 
we may equate to zero the first differential coefficient of the 
weight with regard to the pressure p a , assuming p 1 to be 
constant. The variable term is 

V, j V A J 

and the result of equating to zero the differential coefficient 
of this expression with regard to p a is 

— = (-7— r 7 = 0.5274, 

a number which is somewhat less than Fliegner's experimen- 
tal ratio. 

There is no algebraic maximum to the velocity of flow as 
calculated by equation (208), but the kinetic theory of gases 
gives a theoretical limit to the velocity, for which the follow- 
ing statement is given by Joule. * 

Maximum Velocity of Flow. — According to the kinetic 
theory of gases, the pressure of a gas on the walls of the con- 
taining vessel is due to the impact of the molecules of the 
gas. To estimate the mean velocity of the molecules Joule 

* Memoir Phil. Soc, vol. ix, p. 107. 



1 56 THERMODYNAMICS OF THE STEAM-ENGINE. 

proceeds in the following manner: The weight of one cubic 

metre of gas is — , and the pressure which it exerts on each of 
v 

the six sides of a cubical vessel containing it is p. Suppose 
that the weight — of the gas to be divided into three equal 

portions, one of which oscillates between each pair of faces of 
the cube and produces the pressure by impact, first on one 
and then on the other of the pair. Now if a body have a 
velocity equal to g it will be brought to rest by a force equal 
to its weight acting on it for one second; and that force act- 
ing for two seconds will bring it to rest and then impart to it 
the same velocity in the opposite direction. In two seconds 
there will be g impacts on each of the pair of faces, and it 
will be assumed that the effect of the impacts is equal to that 

of a pressure equal to kilograms on each face; that is, on 

one square metre. The pressure will vary as the square of 
the velocity, since both the force required to reverse the 
velocity and the number of impacts increase with the velocity. 
Finally, Joule makes 

1 



3gv 

in which u is the mean velocity of the molecules of the gas. 
This may be written 



u 



^3 



= Vgpv = VgRT. 



From this discussion Fliegner assumes that the maximum 
velocity of flow of gas through an orifice by the kinetic theory 
of gases is 

V max = V^RT= 16.9/7; 



FLOW OF FLUIDS. 157 

P 
for French units. But his limiting ratio of pressures — = 

0.5767 inserted in equation (208) gives 

V max = 1 7. 1 VT r 

Flow of Saturated Vapor. — For a mixture of a liquid 
and its vapor equation (134) gives 

so that equation (202) gives for the adiabatic flow from a 
receptacle in which the initial velocity is zero 

V 3 1 

— = 34-fo — ?i + *iA - *»Pa) +A^i ~A*V • ( 2I 3) 

Substituting for v 1 and v % from 

V = XU -\- <T y 

A — =g l — g 9 + x 1 p l — * a p a + Ap x xjt x -Ap^x.u, + ^ a{p, —p % ). 

But 

p -f Apu = r ; 

The last term of the right-hand member is small, and fre- 
quently can be omitted. 

The value of x^ can be determined by the equation 

1 + #i = -^f~ + # 9 ; 



1 -1 T 



or, if the proper tables are lacking, we may use the approxi- 
mate form 

-jt- = ~y~ + c log,--- 

■* a -* 1 ■* a 



158 THERMODYNAMICS OF THE STEAM-ENGINE. 

It is necessary to remember that while the tables com- 
monly give the pressure in pounds on the square inch, or in 
atmospheres, etc., p 1 and p 9 in the last term of equation (214) 
are the specific pressures; that is, the pressures in pounds on 
the square foot, or kilograms on the square metre. 

The weight of fluid that will pass through an orifice hav- 
ing an area of a square metres or square feet may be calcu- 
lated by the formula 

aV 
xji^ + a \ 5) 

The equations deduced are applicable to all possible mix- 
tures of liquid and vapor, including dry saturated steam and 
pure hot water. In the first place steam will be condensed 
in the tube, and in the second water will be evaporated. 

If steam blows out of an orifice into the air, or into a large 
receptacle, and comes to rest, the energy of motion will be 
turned into heat and will superheat the steam. Steam blow- 
ing into the air will be wet near the orifice, superheated at a 
little distance, and if the air is cool will show as a cloud of 
mist further from the orifice. 

Rankine's Equations. — After an investigation of the 
experiments made by Mr. R. D. Napier on the flow of steam 
Rankine concludes that the pressure in the orifice is never 
less than the pressure which gives the maximum weight of 
discharge, and that the discharge in pounds per second may 
be calculated by the following empirical equations: 

^ 5 . A . 

A = or > -p a , w = a— , 

v, / 5„ AS 3(A -A) ) i 
A < —p a » w = a — 1 c 

in which p x is the pressure in the reservoir, p a is the pressure 
of the atmosphere, both in pounds on the square inch, and a 
is the area in square inches. 



FLOW OF FLUIDS. 



1 59 



The error of these equations is liable to be about two per 
cent; but the flow through a given orifice may be known more 
closely if tests are made on it at or near the pressure during 
the flow, and a special constant is found for that orifice. 

Experiments on Flow of Steam. — The results of tests 
made on the flow of steam through orifices or short tubes 
with well-rounded entrances, by Mr. W. H. Kunhardt * in 
the laboratories of the Massachusetts Institute of Technology, 
are given in the following table: 

FLOW OF STEAM THROUGH SHORT TUBES WITH ROUNDED 

ENTRANCES. 

Diameters 0.25 of an inch. 









Pressure above A.t 
mosphere, Pounds 




Ratio of 
Absolute 







Flow in Pounds 
per Hour. 






OS 




per Square 


Inch. 




Pressures. 






3 




OJ 










u 

a 




a . 


« cj 








O* 




















W 




C 
1— < 

V 
JO 

3 


en 
V 
4-1 

3 

a 


3 


3 


c 

CJ 
•3.Q 


en 

-a 

li 

, c 
u 

13^3 


u 
u 

3 
</> 

ai en 


v J- • 

P*H 


4; OJ 

u 
a; .c 


3-3 1 

Is: 


c 




n C « 









M-H 



■5 

Si) 

c 

0) 


£ 

c 


3 


H 

X! 

1) 

> 
O 

x> 


"1! 


oh 

"c5 
««5 


v *-" • 

J- OJ ? 

tf3 -J rtj 

gH.o 


aj O <U 

* J > 

O 

*- u re 

3« 

"A" 

4J 3 


<u <u 


^ 


vxi 
— be 


u °.2 
« Ere 

3 HI 3 

0.3. a- 


•Q 3 
re <" 

"re-* 


. 

CJ ^ 

w 

S£ S 
«.2 
■- 




J 


P 


< 


m 


<J 


PQ 


a. 


P* 


H 


Ph 


ffl 


U 


u 


U 


I 


i-5 


SO 


74.1 


14.8 


41.2 


14.7 


0.332 


0.630 


126.2 


I .2 


221 ,0 


217.0 


224 


1. 018 


2 


'* 


30 


71.0 


13.2 


39-6 


14.8 


0.326 


0.634 


138.7 


1-5 


213.0 


207.8 


215 


1.025 


3 




20 


72.6 


19.7 


40 6 


14.7 


0.394 


0.634 


141. 4 


0.5 


216.0 


211. 4 


220 


1.022 


4 


fci 


20 


75-9 


20.4 


42.6 


14.7 


0.387 


0.632 


139.8 


0.7 


228.0 


219.3 


227 


1 .040 


5 


' 


20 


71.9 


24-5 


40.6 


x 4-7 


0.454 


0.638 


140.6 


0.7 


213.0 


209.7 


218 


i .016 


6 


0.5 


30 


72.8 


14.8 


39.0 


14.8 


0.338 


0.614 


138.7 


0.3 


225.0 


213.6 


221 


1-053 


7 




20 


72 . 1 


20.4 


38.8 


14.8 


0.405 


0.617 


142. 2 


0.5 


223.5 


2X1. 7 


219 


0.056 


8 


■' 


30 


72.6 


24.7 


39.0 


14.8 


0.452 


0.616 


144.0 


OS 


223.0 


213. 1 


220 


1 .046 


9 


" 


3° 


73-i 


29,9 


39 - 2 


14 8 


0.509 


615 


145.2 


O.5 


225.5 


213.O 


222 


1-054 


IQ 


0.25 


3° 


72.6 


24.8 


36.1 


14.9 


0.454 


0.583 


143.8 


O.4 


225.0 


213.5 


220 


1.054 


II 




30 


72.6 


19.9 


36.1 


14.9 


0.398 


0.583 


141 .0 


O.4 


225.0 


2*3-5 


220 


1.054 


12 


" 


3o 


72.7 


14.9 


36.2 


14.8 


0.339 


0.583 


140.5 


O.4 


227.0 


213.0 


220 


1 .066 


13 


" 


30 


126.3 


27.8 


69.0 


14.7 


0.295 


0.594 


iSS-o 


0.5 


358.8 


338.9 


355 


1.058 


14 




30 


125.0 


40.8 


67.9 


14.7 


0.398 


0.598 


157-0 


0.2 


355 -o 


334-8 


352 


1.060 



The following table gives the results of some experiments 
on the flow of steam through an orifice 0.25 of an inch in 
diameter, in a thin plate, made by Mr. G. P. Aborn f in the 
laboratories of the Massachusetts Institute of Technology: 



* Transactions Am. Soc. Mech. Engs. t vol. xi, p. 187. 
\ Thesis, 1886. 



lOO 



THERMODYNAMICS OF THE STEAM-ENGINE. 



FLOW OF STEAM THROUGH AN ORIFICE. 



Number of 


Higher 


Difference of 


Flow in Pounds 
per Hour by 


Experiment. 


Pressure. 


Pressure. 








Experiment. 


I 


71.8 


O.92 


29.7 


2 


7i-5 


1.85 


43-1 


3 


71.9 


2.79 


52.6 


4 


71.6 


3-89 


67.6 


5 


71.9 


5-55 


77.6 


6 


71.8 


6.50 


84.2 


7 


71.7 


8.07 


91.8 


8 


72.9 


9-23 


93-9 


9 


72.5 


12.8 


no. 3 


IO 


73-7 


15.9 


124.9 


ii 


72.7 


21. 1 


I4I-5 


12 


74-2 


27.0 


156.8 


13 . 


71.9 


33-7 


166.3 


14 


74-3 


41 


180.7 


15 


72.7 


49.2 


187.7 


76 


72.9 


57-0 


195.8 


17 


73-7 


64.4 


196.9 


18 


72.0 


68.4 


197.8 



Flow of Superheated Steam. — The form of the equation 
for the change of intrinsic energy of superheated steam is the 
same as for a perfect gas, i.e., 

1 3 ~~ k - 1 k - 1 

and consequently the equations for the velocity of flow and 
the weight of the discharge can be deduced in much the same 
form as for a perfect gas, provided that the steam remains 
superheated. Though there are no experiments on the flow 
of superheated steam under such conditions, it is probable 
that the ratio of the pressures in the orifice and in the reservoir 
is something between 0.57 and 0.6. 

But steam is seldom sufficiently superheated to avoid con- 
densation during adiabatic flow through an orifice. If the 
steam becomes moist in the orifice, then the intrinsic energy 
at the initial condition must be found from equation (179), 
which may be written 



k- 1 



M^4)+W h/0, 



A 



. (216) 



FLOW OF FLUIDS. l6l 

while the intrinsic energy at the pressure in the orifice is given 
by the equation for saturated steam, 

m 

£*F=jf(&+X*P a ), (217) 

so that the equation for the velocity of flow becomes 



Tg = A I- i* 1 +Z^ ~ q% + Pl ~ x ^ +AVl ~ A ^ 2 ' ^ 2l8 ) 



in which v 1 is to be calculated from the temperature T x and 
the pressure ^ by aid of equation (176) or (177), while z/ a is 
to be found from the equation 

v a = x,u 2 + a. 

The value of x^ for the final condition can be determined 
by aid of equation (185), 

T 1 X T 

-jr + d.+cp log, ~Y = --T + * 

Finally, the weight discharged per second may be found 
by the equation 



aV 

w = — . 



In this work the most expeditious way is to make numer- 
ical calculations by the several equations without attempting 
any further algebraic reduction. The pressure in the orifice 
will be very nearly 0.6 of the pressure p 1 in the reservoir, 
provided that p z is more than ^ the pressure of the atmosphere 
into which discharge takes place. 



1 62 THERMODYNAMICS OF THE STEAM-ENGINE. 

EXAMPLES. 

1. Find the velocity of flow of air from the pressure of 
6 atmospheres in a reservoir to the pressure of 5 atmospheres 
in the throat of the orifice; also from 5 to 4 atmospheres, 
from 4 to 3, and from 3 to 2, the initial temperature in each 
case being 30° C. Ans. 175.8, 193.9, 219.2,258.0 metres 
per sec. 

2. Find the weight of air per second that will be dis- 
charged from an orifice 1 inch in diameter, from a reservoir 
having the temperature 6o° F. and a pressure of 150 pounds 
per square inch, into the atmosphere. Ans. 2.736 lbs. 

3. Find the weight of saturated steam per second dis- 
charged through an orifice 1 inch in diameter, from a boiler 
having the gauge-pressure 60 pounds, into the atmosphere. 
Find also for the following values of x, 0.9, 0.8, 0.6, 0.5, 
0.4, 0.2, and for hot water. Ans. 0.850 lbs.; 0.893, 0.943, 
1.077, 1. 169, 1.289, 1.710,3.064. 

4. Find the velocity of flow of superheated steam with 
the initial temperature 360 F. and initial pressure 100 pounds 
absolute, when the pressure in the throat of the orifice is 60 
pounds absolute. Ans. 1400 ft. per sec. 

5. In example 4 find the weight per second discharged 
through an orifice 1 inch in diameter. Ans. 1.09 lbs. 



CHAPTER IX. 
INJECTORS. 

An injector is an instrument by means of which a jet of 
steam acting on a stream of water with which it mingles, and 
by which it is condensed, can impart to the resultant jet of 
water a sufficient velocity to overcome a pressure that may 
be equal to or greater than the initial pressure of the steam. 
Thus, steam from a boiler may force feed-water into the same 
boiler, or into a boiler having a higher pressure. The 
mechanical energy of the jet of water is derived from the heat 
energy yielded by the condensation of the steam-jet. There 
is no reason why an injector cannot be made to work with any 
volatile liquid and its vapor, if occasion may arise for doing 
so; but in practice it is used only for forcing water. An 
essential feature in the action of an injector is the condensa- 
tion of the steam by the water forced; other instruments using 
jets without condensation, like the water-injector in which a 
small stream at high velocity forces a large stream with a low 
velocity, differ essentially from the steam-injector. 

Method of Working". — A very simple form of injector is 

shown by Fig. 31, consisting of three essential parts, a the 

steam-nozzle, b the combining-tube, and c the delivery -tube. 

Steam is supplied to the injector through a pipe connected 

at d\ water is supplied through a pipe at /*, and the injector 

forces water out through the pipe at e. The steam-pipe must 

have on it a valve for starting and regulating the injector, and 

the delivery-pipe leading to the boiler must have on it a 

163 



164 



THERMODYNAMICS OF THE STEAM-ENGINE. 



check-valve to prevent water from the boiler from flowing 
back through the injector when it is not working. The 
water-supply pipe commonly has a valve for regulating the 
flow of water into the injector. 

This injector, known as a non- lifting injector, has the 
water-reservoir set high enough so that water will flow into 
the injector through the influence of gravity. A lifting 




Fig. 31. 



injector has a special device for making a vacuum to draw 
water from a reservoir below the injector, which will be 
described later. 

To start the injector shown by Fig. 31, the steam-valve is 
first opened slightly to blow out any water that may have 
gathered above the valve, through the overflow, since it is 
essential to have dry steam for starting. The steam-valve is 
then closed, and the water-valve is opened wide. As soon 
as water appears at the overflow between the combining-tube 
and the delivery-tube the steam-valve is opened wide, and the 
jet of steam from the steam-nozzle mingles with and is con- 
densed by the water and imparts to it a high velocity, so that 
it passes across the overflow space between the combining- 
tube and the delivery-tube and passes into the boiler. When 
the injector is working a vacuum is formed at the space 



INJECTORS. 165 

between the combining and delivery tubes, and the valve at 
the overflow then closes and excludes air which would mingle 
with the water and might interfere with the action of the 
injector. 

Theory of the Injector. — The two fundamental equa- 
tions of the theory of the injector are deduced from the prin- 
ciples of the conservation of energy and the conservation of 
momenta. 

The heat energy in one pound of steam at the absolute 
pressure p x in the steam-pipe is 



^"(*i r i + *•)» 



where r x and q 1 are the heat of vaporization and heat of the 

liquid corresponding to the pressure^; — is the mechanical 

equivalent of heat (778 foot-pounds), and x x is the quality of 
the steam; if there is two per cent of moisture in the steam, 
then x x is 0.98. 

Suppose that the water entering the injector has the tem- 
perature t 3 , and that its velocity where it mingles with the 
steam is VJ ; then its heat energy per pound is 



A 



1 



and its kinetic energy is 



V n 



where q % is the heat of the liquid at t 3 , and g is the accelera- 
tion due to gravity (32.2 feet). 

If the water forced by the injector has the temperature l iy 



l66 THERMODYNAMICS OF THE STEAM-ENGINE. 

and if the velocity of the water in the smallest section of the 
delivery-tube is V W} then the heat energy per pound is 



i 



and the kinetic energy is 



A 



V m 



zg 



Let each pound of steam draw into the injector y pounds 
of water; then, since the steam is condensed and forced 
through the delivery-tube with the water, there will be I -\- y 
pounds delivered for each pound of steam. Equating the sum 
of the heat and kinetic energies of the entering steam and 
water to the sum of the energies in the water forced from the 
injector, we have 

i / i V /2 \ / i V 2 \ 

A(*s l + g l )+y[-jg, + — -) =(i+y){-j<? t + ~). (219) 

The terms depending on the velocities VJ and V w are 
never large and can commonly be neglected. Thus, for a 
lifting-injector the pressure causing water to enter the injector 
is always less than the pressure of the atmosphere and for a 
non-lifting injector it is only a little more. If the pressure 
of the atmosphere is 14.7 pounds per square inch and if there 
is a perfect vacuum in the injector, then by equation (204), 
the heat equivalent of the term depending on VJ, assuming 
y to be 15, will be 

y~-> ='5X^X 144(14.7 - 0)g^ = 0.6 B. T. U. 

If the injector delivers water against a boiler-pressure of 
150 pounds by the gauge or 164.7 pounds absolute, then the 



JNJECTORS. 167 

heat equivalent of the term depending on V w will be 

AVJ 16 X 144 X 164.7 

(y + I) = ^~-p — = 7-8 B. T. U. 

KJ ' } 2g 778 X 62.4 ' 

In both calculations the pressure of the water-jet in the 
smallest section of the combining tube is assumed to be small 
enough to be neglected. The determination of the term 
depending on V w comes from the consideration that V w must 
be greater than the velocity with which cold water would flow 
out under the influence of the boiler-pressure. 

Now since r 1 is always greater than 800, the term depend- 
ing on V w is about one per cent of the total left-hand mem- 
ber of equation (219), and the term depending on VJ is less 
than one tenth of a per cent. For practical calculations we 
may neglect both, reducing equation (219) to 

*,', + ?.-?, (220) 

Equation (220) may be applied to any kind of injector, 
including double injectors which have two steam-nozzles. 

For example, if dry steam is supplied tc the injector at 
120 pounds by the gauge or 134.7 pounds absolute, if the 
supply-temperature of the water is 65 F., and if the delivery- 
temperature is 165 F., then the water pumped per pound of 
steam is 

r* + 4i - <1* 867.4 +32T. 2 - 133 .4 • 

y = = = 10.5 pounds. 

q<-q 3 133-4- 33-12 v 

The momentum of one pound of steam issuing from the 
steam-nozzle with the velocity V s is V s -s- g\ the momentum 
of y pounds of water entering the combining-tube with the 
velocity VJ is yVJ -f- g\ and the momentum of 1 -\- y pounds 
of water at the smallest section of the deliverv-tube is 



1 68 THERMODYNAMICS OF THE ^STEAM-ENGINE. 

(I -f- y) V w -r- g. Equating the sum of the momenta of 
water and steam before mingling to the momentum of the 
combined water and steam in the delivery-tube, 

V.+jrVJ = {i+y)V m (221) 

This equation can be used to calculate any one of the 
velocities provided the other two can be determined inde- 
pendently. Unfortunately there is much uncertainty about 
all of the velocities so that the proper sizes of the orifices and 
of the forms and proportions of the several members of an 
injector have been determined mainly by experiment. The 
best exposition of this matter is given by Mr. Strickland 
Kneass,* who has made many experiments for Wm. Sellers 
& Co. The practical part of what follows is largely drawn 
from his work. 

Velocity of the Steam-jet. — Equation (214) gives 

Vs ~ I ~aS XxY ' ~~ x * r * + ?» ~ ?«) ) » • • ( 222 ) 

where r 1 and q x are the heat of vaporization and the heat of 
the liquid of the supply of steam at the pressure/^ and r 2 
and q^ are corresponding quantities at the pressure / 2 for that 
section of the tube for which the velocity is calculated ; x x is 
the quality of the steam at the pressure p x (usually 0.98 to 
unity) and x^ is the quality at the pressure / 2 to be calculated 
by aid of the equation 



+ l = i£ + e r 



T ' ' T 

Here T x and T^ are the absolute temperatures corresponding 
to the pressures /, and / 2 , and B x and #„ are the entropies of 

* Practice and Theory of the Injector, J. Wiley & Sons. 



INJECTORS. 169 

the liquid at the same pressures. Of course -7 is the me- 

chanical equivalent of heat and g is the acceleration due to 
gravity. 

In the discussion of the flow of steam it appeared that the 
pressure in an orifice with rounded approach is about j\ of 
the absolute pressure of the steam in the pipe leading to the 
orifice, and that the quantity of steam discharged is not 
affected by the pressure against which the discharge takes 
place, provided the latter is not more than half the pressure 
causing the flow. This principle may be applied with fair 
approximation to the injector, so far as the calculation of the 
amount of steam used by an injector is concerned; conversely 
the same principle can be used for calculating the size of the 
orifice in the steam-nozzle. If preferred, Rankine's equation 
(page 158) can be used for this purpose. But it is pointed 
out by Mr. Kneass that by flaring or expanding the steam- 
nozzle the pressure in it may be reduced, and consequently 
the velocity of the steam discharged may be very much 
increased. Experiments on a nozzle of a Sellers' injector 
showed that the pressure at the exit when the injector was 
working was a little less than the pressure of the atmosphere 
for various pressures from 60 to 120 pounds by the gauge. 
Applying to a special case, we have, 

For example, the velocity of discharge from a straight 
orifice under the pressure of 120 pounds by the gauge or 
134.7 pounds absolute is 



V* = J ^"(^^1 + *f* + ft - ft) j 



= {2X 32.2 X 778(867.4—0.967x895.1 + 321.2 — 282.1)}* 
= 1430 feet per second, 

having for x a 



I/O THERMODYNAMICS OF THE STEAM-ENGINE. 

= O.967O, 

provided that/ a = 0.6/, = 80.8 pounds absolute. 

If, however, the pressure at the exit of an expanded 
nozzle is 14.7 pounds absolute, then 



x n = 



672.7/867.4 , \ 

56^s(sl^7 + a5025 - a3I27 i = a8775> 



and 



F; = )2X 32.2X778(867.4 — 0.8711 X 965.8+321.2— 180.8)}* 
= 2830 feet per second, 

which is nearly twice that just calculated for the velocity at 
the smallest section of the steam-nozzle. 

Velocity of Entering Water. — The velocity of the water 
in the combining-tube where it mingles with the steam 
depends on (a) the lift or head from the reservoir to the 
injector, (b) the pressure (or vacuum) in the combining-tube, 
and (c) on the resistance which the water experiences from 
friction and eddies in the pipe, valves, and passages of the 
injector. The first of these can be measured directly for any 
given case; for example, where a test is made on an injector. 
In determining the proportions of an injector it is safe to 
assume that there is neither lift nor head for a non-lifting 
injector, and that the lift for a lifting-injector is as large as 
can be obtained with certainty in practice. The lift for an 
injector is usually moderate, and seldom if ever exceeds 
20 feet. 

The vacuum in the combining-tube may amount to 22 or 
24 inches of mercury, corresponding to 25 or 27 feet of water; 
that is, the absolute pressure may be 3 or 4 pounds per square. 



INJECTORS. \yi 

inch. The vacuum after the steam and water are combined 
appears to be limited by the temperature of the water; 
thus, if the temperature is 165 F., the absolute pressure 
cannot be less than 5.3 pounds. But the final temperature 
is taken in the delivery-pipe after the water and condensed 
steam are well mixed and are moving with a moderate 
velocity. 

The resistance of friction in the pipes, valves, and passages 
of injectors has never been determined; since the velocity is 
high the resistance must be considerable. 

If we assume the greatest vacuum to correspond to 27 feet 
of water, the maximum velocity of the water entering the 
combining-tube will not exceed 



V2gh = V 2 X 32.2 X 27 = 42 feet. 

If, on the contrary, the effective head producing velocity 
is as small as 5 feet, the corresponding velocity will be 



V2 X 32.2 X 5 = 18 feet. 

It cannot be far from the truth to assume that the velocity 
of the water entering the combining-tube is between 20 and 
40 feet per second. 

Velocity in the Delivery-tube. — The velocity of the 
water in the smallest section of the delivery-tube may be 
estimated in two ways: in the first place it must be greater 
than the velocity of cold water flowing out under the pressure 
in the boiler, and in the second place it may be calculated by 
aid of equation (221), provided that the velocities of the 
entering steam and water are determined or assumed. 

For example, let it be assumed that the pressure of the 
steam in the boiler is 120 pounds by the gauge, and that, 
as calculated on page 167, each pound of steam delivers 10.5 
pounds of water from the reservoir to the boiler. As there 
is a good vacuum in the injector we may assume that the 



I7 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

pressure to be overcome is 132 pounds per square inch, corre- 
sponding to a head of 

132 X 144 _ , . 

- = 305 feet. 

62.4 J D 

Now the velocity of water flowing under the head of 305 
feet is 



V2gh = V2 X 32.2 X 305 = 140 feet per second. 

The velocity of steam flowing from a pressure of 120 
pounds by the gauge through a diverging-tube with the pres- 
sure equal to that of the atmosphere at the exit has been 
calculated to be 2830 feet per second. Assuming the velocity 
of the water entering the combining-tube to be 20 feet, then 
by equation (221) we have in this case 

Vw = Z^qLL = 2830 +10.5 x 20 = 266 
1+7 1 + 10.5 

If the steam is assumed to flow through a converging- 
nozzle then the velocity at 120 pounds boiler-pressure will 
be 1430 feet per second, and the velocity of the water-jet 
becomes 

1430+ 10.5 X20 

V w = — = 143 feet. 

1 -f- 10.5 ^° 

Since there is considerable resistance due to friction and 
to sudden change of velocity in leaving the delivery-tube, this 
last velocity is probably insufficient. But if the entrance 
velocity is assumed to be 30 feet, then the calculation gives 
151 feet for the velocity in the delivery-tube, which is prob- 
ably enough to make the injector work. 

Sizes of the Orifices. — From direct experiments on in- 
jectors it appears that the quantity of steam delivered by the 
steam-nozzle can be calculated in all cases by the method for 



INJECTORS. 173 

the flow of steam through a straight orifice, assuming the 
pressure in the orifice to be -^ of the absolute pressure above 
the orifice. 

Now each pound of steam forces y pounds of water from 
the reservoir to the boiler; consequently if w pounds are 
drawn from the reservoir per second the injector will use 
w -— y pounds of steam per second. 

The specific volume of the mixture of water and steam in 
the smallest section of the steam-nozzle is 

v, = x,u 2 -f a, 

where x^ is the quality, u^ is the increase of volume due to 
vaporization, and a is the specific volume of the water. The 
volume of steam discharged per second is 

y '. 

and the area of the orifice is 



a, = 



yV s 



(223) 



where V s is the velocity at the smallest section. 

For example, for a flow from 134.7 pounds absolute to 
80.8 pounds absolute x t is 0.9670 and V s is 1430 feet, as 
found on page 169. Again, for an increase of temperature 
from 65 F. to 165 F. the water per pound of steam is 10.5. 
Calculating the specific volume at 80.8 pounds, we have 

v 2 — xji^-\- <r = 0.9670(5.375 —0.016)+ 0.016 = 5. 198 cubic feet. 

If the injector is required to deliver 1200 gallons an hour, 
or 

1200 X 231 X 62.4 



1728 X 60 X 60 



= 2.78 



174 THERMODYNAMICS OF THE STEAM-ENGINE. 

pounds per second, the area of the steam-nozzle must be 

wv % 2.78 X 5-I98 , - 

a. = — — = = 0.000902 square feet. 

yV s 10.5 X 1430 

The corresponding diameter is 0.420 of an inch, or 10.6 mil- 
limetres. 

In trying to determine the size of the orifice in the 
delivery-tube we meet with two serious difficulties: we do not 
know the velocity of the steam in the smallest section of the 
delivery-tube, and we do not know the condition of the fluid 
at that place. It has been assumed that the steam is entirely 
condensed by the water in the combining-tube before reach- 
ing the delivery-tube, but usually there are small bubbles of 
uncondensed steam still mingled with the water, so that the 
actual density of the heterogeneous mixture is from 0.6 to 
0.9 that of water. Since the pressure at the entrance to the 
delivery-tube is small, the specific volume of the steam is very 
large, and a fraction of a per cent of steam is enough to reduce 
the density of the steam to one-half. Even if the steam is 
entirely condensed, the air carried by the water from the 
reservoir is enough to sensibly reduce the density at the low 
pressure (or vacuum) found at the entrance to the delivery- 
tube. 

If V w is the probable velocity of the jet at the smallest 
section of the delivery-tube, and if 6 is the density of the 
fluid, then the area of the orifice in square feet is 

w i l +y) * v 

* = ~MT' ••••••• ( 22 4) 

for each pound of steam mingles with and is condensed by y 
pounds of water and passes with that water through the 
delivery-tube; w f as before, is the number of pounds of water 
drawn from the reservoir per second. 

For example, let it be assumed that the actual velocity in 






INJECTORS, 175 

the delivery-tube to overcome a boiler-pressure of 120 pounds 
by the gauge is 150 feet per second, and that the density of 
the jet is about 0.9 that of water; then with the value of 
w = 2.78 and y = 10.5, we have 

w(\ -\-y) 2.78 X 1 1.5 

a w = JT , = — — — — -7 — = 0.000361 sq. ft. 

V w Sy 150X0.9X62.4X10.5 ° M 

The corresponding diameter is 0.257 of an inch, or 6.5 mil- 
limetres. 

Steam-nozzle. — The entrance to the steam-nozzle should 
be well rounded to avoid eddies or reduction of pressure as 
the steam approaches; in some injectors, as the Sellers' 
injector, Fig. 32, the valve controlling the steam supply is 
placed near the entrance to the nozzle, but the bevelled 
valve-seat will not interfere with the flow when the valve 
is open. 

It has already been pointed out that the steam-nozzle may 
advantageously be made to expand or flare from the smallest 
section to the exit. The length from that section to the end 
may be between two and three times the diameter at that 
section. 

Consider the case of a steam-nozzle supplied with steam 
at 120 pounds boiler-pressure: it has been found that the 
velocity at the smallest section, on the assumption that the 
pressure is then 80.8 pounds, is 1430 feet per second, and that 
the specific volume is 5.198 cubic feet. If the pressure is 
reduced to 14.7 pounds, that is, to atmospheric pressure, at 
the exit, the velocity becomes 2830 feet per second, the 
quality being x^ = 0.8775. The specific volume is conse- 
quently 

v^ = x^u 2 + & — 0.8775(26.60 — 0.016) + 0.016 = 23.34 cu. ft. 

The areas will be directly as the specific volumes and 
inversely as the velocities, so that for this case we shall have 



176 THERMO D YNAMICS OF THE STEAM-ENGINE. 

the ratio of the areas 

5.198: 23.34] 

\ = I : 2.27 

2830 : 1430 [ ' 

and the ratio of the diameters will be 



Vi . V2.27 = 1 : 1.5. 

Combining-tube. — There is great diversity with different 
injectors in the form and proportions of the combining-tube. 
It is always made in the form of a hollow converging 
cone, straight or curved. The overflow is commonly con- 
nected to a space between the combining-tube and the 
delivery-tube; it is, however, sometimes placed beyond the 
delivery-tube, as in the Sellers' injector, Fig. 32. In the 
latter case the combining and delivery tubes may form one 
continuous piece, as is seen in the double injector shown by 

Fig. 33- 

The Delivery-tube. — This tube should be gradually en- 
larged from its smallest diameter to the exit in order that the 
water in it may gradually lose velocity and be less affected by 
the sudden change of velocity where this tube connects to the 
pipe leading to the boiler. 

It is the custom to rate injectors by the size of the 
delivery-tube; thus a No. 6 injector may have a diameter of 
6 mm. at the smallest section of the delivery-tube. 

Mr. Kneass found that a delivery-tube cut off short at the 
smallest section would deliver water against 35 pounds pres- 
sure only, without overflowing; the steam-pressure being 65 
pounds. A cylindrical tube four times as long as the internal 
diameter, under the same conditions would deliver only 
against 24 pounds. A tube with a rapid flare delivered 
against 62 pounds, and a gradually enlarged tube delivered 
against 93 pounds. 

If the delivery-tube is assumed to be filled with water 



INJECTORS, 177 

without any admixture of steam or air, then the relative 
velocities at different sections may be assumed to be propor- 
tional to the corresponding areas. This gives a method 
of tracing the change of velocity of the water in the tube 
from its smallest diameter to the exit. 

A sudden change in the velocity is very undesirable, as 
at the point where the change occurs the tube is worn and 
roughened, especially if there are solid impurities in the water. 
It has been proposed to make the form of the tube such that 
the change of velocity shall be uniform until the pressure has 
fallen to that in the delivery-pipe; but this idea is found to be 
impracticable, as it leads to very long tubes with a very wide 
flare at the end. 

Efficiency of the Injector. — The injector is used for 
feeding boilers, and for little else. Since the heat drawn from 
the boiler is returned to the boiler again, save the very small 
part which is changed into mechanical energy, it appears as 
though the efficiency was perfect, and that one injector is 
as good as another provided that it works with certainty. 
We may almost consider the injector to act as a feed-water 
heater, treating the pumping in of feed-water as incidental. 
It has already been pointed out on page 167 that the kinetic 
energy of the jet in the delivery-tube is less than one per cent 
of the energy due to the condensation of the steam. On this 
account the injector is used wherever cold water must be 
forced into a boiler, as on a locomotive, or when sea-water is 
supplied to a marine boiler. Considering only the advantage 
of supplying hot water to the boiler, it almost seems as 
though the more steam an injector uses the better it is. 
Such a view is erroneous, as it is often desirable to supply 
water without immediately reducing the steam-pressure and 
then it is necessary to use as little steam as may be. It is, 
however, true that simplicity of construction and certainty of 
action are of the first importance in injectors. 

Lifting Injector. — The injector described at the begin- 



i;8 



THERMODYNAMICS OF THE STEAM-ENGINE. 



ing of this chapter was placed so that water from the reservoir 
would run in under the influence of gravity. When the 




injector is placed higher than the reservoir a special device is 
provided for lifting the water to start the injector. Thus in 
the Sellers' injector, Fig. 32, there is a long tube which pro- 



INJECTORS 1 79 

trudes well into the combining-tube when the valves w and x 
are both closed. When the rod B is drawn back a little by 
aid of the lever H the valve w is opened, admitting steam 
through a side orifice to the tube mentioned. Steam from 
this tube drives out the air in the injector through the over- 
flow, and water flows up into the vacuum thus formed, and is 
itself forced out at the overflow. The starting-lever H is then 
drawn as far back as it will go, opening the valve x and sup- 
plying steam to the steam-nozzle. This steam mingles with 
and is condensed by the water and imparts to the water 
sufficient velocity to overcome the boiler-pressure. Just as 
the lever H reaches its extreme position it closes the overflow 
valve K through the rod L and the crank at R. 

Since lifting-injectors may be supplied with water under 
a head, and since a non-lifting injector when started will lift 
water from a reservoir below it, or may even start with a 
small lift, the distinction between them is not fundamental. 

Double Injectors. — The double injector illustrated by 
Fig. 33, which represents the Korting injector, consists of 
two complete injectors, one of which draws water from the 
reservoir and delivers it to the second, which in turn delivers 
the water to the boiler. To start this injector the handle A 
is drawn back to the position B and opens the valve supply- 
ing steam to the lifting-injector. The proper sequence in 
opening the valves is secured by the simple device of using a 
loose lever for joining both to the valve-spindle; for under 
steam-pressure the smaller will open first, and when it is open 
the larger will move. The steam-nozzle of the lifter has a 
good deal of flare, which tends to form a good vacuum. The 
lifter first delivers water out at the overflow with the starting- 
lever at B\ then that lever is pulled as far as it will go, open- 
ing the valve for the second injector or forcer, and closing 
both overflow valves. 

Automatic Injectors. — In the discussions of injectors 
thus far given it has been assumed that they work at full 
capacity, but as an injector must be able to bring the water- 



i8o 



THERMODYNAMICS OF THE STEAM-ENGINE. 



level in a boiler up promptly to the proper height, it will 
have much more than the capacity needed for feeding the 
boiler steadily. Any injector may be made to work at a 
reduced capacity by simply reducing the opening of the 
steam-valve, but the limit of its action is soon reached. The 




Fig. 33. 

limit may be extended somewhat by partially closing the 
water-supply valve and so limiting the water-supply. 

The original Giffard injector had a movable steam-nozzle 
to control the thickness of the sheet of water mingling with 
the steam, and also had a long conical valve thrust into the 
steam-nozzle by which the effective area of the steam-jet 
could be regulated. Thus both water and steam passages 
could be controlled without changing the pressures under 
which they were supplied, and the injector could be regulated 
to work through a wide range of pressures and capacities. 
The main objection was that the injector was regulated by 
hand and required almost constant attention. 

In the Sellers injector, Fig. 32, the regulation of the 
steam-supply by a long cone thrust through the steam-nozzle 
is retained, but the supply of water is regulated by a movable 



INJECTORS. l8l 

combining-tube, which is guided at each end and is free to 
move forwards and backwards. At the rear the combining- 
tube is affected by the pressure of the entering water, and in 
front it is subjected to the pressure in the closed space O, 
which is in communication with the overflow space between 
the combining-tube and the delivery-tube, in this injector the 
space is only for producing the regulation of the water-supply 
by the motion of the combining-tube, as the actual overflow 
is beyond the delivery-tube at K. When the injector is 
running at any regular rate the pressures on the front and the 
rear of the combining-tube are nearly equal, and it remains at 
rest. If the supply of steam is decreased by partially closing 
the steam-valve or by reducing the boiler-pressure, the 
velocity of the water in the combining-tube is diminished and 
the pressure in the chamber O increased, so that the combin- 
ing-tube is forced back and the supply of water is regulated 
to correspond with the steam-supply. A contrary action 
ensues if the supply of steam is increased either by raising the 
boiler-pressure or by opening the steam-valve wider. The 
injector is always started at full capacity by pulling the steam- 
valve wide open, as already described ; after it is started the 
steam-supply is regulated at will by the engineer or boiler 
attendant, and the water is automatically adjusted by the 
movable combining-tube, and the injector will require atten- 
tion only when a change of the rate of feeding the boiler is 
required on account of either a change in the draught of steam 
from the boiler, or a change of steam-pressure, for the 
capacity of the injector increases with a rise of pressure. 

The action of this injector is well represented by the fol- 
lowing table of experimental results, furnished by the makers: 

For each pressure of steam noted in column I, the water 
was delivered by the injector into the boiler under approxi- 
mately the same pressure. The delivery was measured by 
observing the indications of a water-meter. The pressures in 
column 8 were obtained by throttling the steam supplied to 
the injector, and observing the pressure at which it ceased to 



182 



THERMOD YNAMICS OF THE STEAM-ENGINE. 



EXPERIMENTS ON A SELLERS INJECTOR. 
(Diam. Water-orifice 6 mm.) 



pplied 

essure 

is de- 

Inch. 


Delivery 
in Cubic Feet per Hour. 


Temperature, 
Fahrenheit Degrees. 


"2* 


eJ) u 

8 « 
■ofe 

TO 


Pressure of Steam su 
to Injector, and Pr 
against which Water 
livered. Lbs. per Sq. 












Pressure of Steam re< 
to deliver Water a 
Pressure in Column 1 


a 

3 
a 


a 

3 

a 

'5 


a£ 

3X 

'SQ 

"* 3 
0.8 

.2 IS 

S3 


u 

ctf 

U 
V 


Delivered Water. 


3 « 
2 * 


a 

a^ 

X > 

< 


a . 

il 

< 


Highest Tempe 
sible of Feed 
Degrees. 


I 


2 


3 


4 


5 


6 


7 


8 


9 


10 


75-3 


63.6 


P.845 


66 


100 


94 


3 


132 


20 


82.4 


61.2 


o-743 


66 


108 


104 


9 


J 34 


30 


94- 2 


56.5 


0.600 


66 


114 


u6 


16 


134 


40 


100. 1 


60.0 


°-599 


66 


120 


123 


22 


132 


50 


108.3 


64.7 


o-597 


66 


124 


125 


27 


131 


60 


116. 5 


63.6 


0.546 


66 


127 


J 33 


34 


130 


70 


124.8 


63.6 


0.510 


67 


130 


142 


40 


130 


80 


133° 


67.1 


0.505 


66 


i34 


144 


46 


131 


90 


i4i-3 


69-5 


0.492 


67 


136 


148 


52 


T 3 2 


100 


147.2 


64.7 


0.456 


66 


140 


»59 


58 


132 


no 


i53-° 


67.1 


0.439 


67 


144 


162 


63 


I32 


120 


156.6 


73-o 


0.466 


67 


148 


162 


69 


134 


130 


161 .2 


74.2 


0.460 


66 


150 


i65 


75 


*3° 


140 


166.0 


78.9 


0.476 


66 » 


153 


166 


81 


I26 


ISC- 


170.7 


70.6 


0.414 


66 


157 


167 


88 


121 



work, each experiment being repeated several times with pre- 
cisely the same results. The temperatures in column 9 were 
obtained by gradually heating the w r ater supplied to the 
injector, and noting the temperature at which it ceased to 
operate, each temperature recorded being checked by several 
repetitions of the experiment. 

A double injector, such as that represented by Fig. 33, is 
to a certain extent automatic, since an increase of steam- 
pressure causes at once an increase in the amount of water 
drawn in by the lifter and an increase in the flow of steam 
through the steam-nozzle of the forcer. Such injectors have 
a wide range of action and can be controlled by regulating 
the valve on the steam-pipe. 



INJECTORS. 



183 



Restarting Injectors. — If the action of any of the injec- 
tors thus far described is interrupted for any reason, it is 
necessary to shut off steam and start the injector anew; 
sometimes the injector has become heated while its action is 
interrupted, and there may be difficulty in starting. To 
overcome this difficulty various steam 

forms of restarting injectors have 
been devised, such as the Gresham 
represented by Fig. 34. This in- 
jector has four jets arranged in line, 

as follows: the steam-nozzle, the P: 

W 
draught-tube, the movable com- l;^>— > 

bining-tube, and the delivery-tube. s 
When the injector is not working 
the movable combining-tube rests 
on the upper end of the tube in 
which it slides, and leaves a free 
passage to the overflow. When 
steam is admitted through the 
steam-nozzle it expels the air in 
the injector, draws in water from 
the reservoir and throws it out at 
the overflow. As soon as the supply of water is established 
so that it can condense the steam, sufficient velocity is im- 
parted to cause the water to pass through the delivery-tube, 
and a vacuum is produced above the combining-tube which 
draws it up and shuts off the passage to the overflow. If the 
action of the injector is temporarily interrupted the combin- 
ing-tube drops down and the instrument is ready to start as 
soon as water is supplied to it. 

Sellers' self-acting injector, represented by Fig. 35, is 
both an automatic and a restarting injector. It is a double 
injector with all the jets in one line; a, b, and c are the 
steam-nozzle, the combining-tube, and the delivery-tube of 
the forcer; the lifter is composed of the annular steam-nozzle 
d, and the annular delivery-tube e, surrounding the nozzle a. 




DELIVERY 

Fig. 34. 



1 84 THERMODYNAMICS OE THE STEAM-ENGINE. 




INJECTORS. 185 

The proportions are such that the lifter can always produce a 
suction in the feed-pipe even when there is a discharge from 
the main steam-nozzle, and it is this fact that establishes the 
restarting feature. When the feed-water rises to the tubes it 
meets the steam from the lifter-nozzle and is forced in a thin 
sheet and with high velocity into the combining-tube of the 
forcer, where it comes in contact with the main steam-jet, 
and mingling with and condensing it, receives a high velocity 
which enables it to pass the overflow orifices and proceed 
through the delivery-tube to the boiler. 

Like any double injector, the lifter and forcer have a con- 
siderable range of action through which the water is adjusted 
to the steam-supply; but there is a further adjustment in this 
injector, for when a good vacuum is established in the space 
surrounding the combining-tube, water can enter through the 
check-valve/", and flowing through the orifices in the combin- 
ing-tube mingles with the jet in it, and is forced with that jet 
into the boiler. 

The steam-valve is seated on the end of the lifter-nozzle, 
and it has a protruding plug which enters the forcer-nozzle. 
When the valve is opened to start the injector, steam is sup- 
plied first to the starter, and soon after, by withdrawing the 
plug, to the forcer. If the steam is dry the starting-lever 
may be moved back promptly; if there is condensed water in 
the steam-pipe, the starting-handle should be moved a little 
way to first open the valve of the lifter, and then it is drawn 
as far back as it will go, as soon as water appears at the over- 
flow. The water-supply may be regulated by the valve g, 
which can be rotated a part of a turn. The minimum delivery 
of the injector is obtained by closing this valve till puffs of 
steam appear at the overflow, and then opening it enough 
to stop the escape of steam. 

When supplied with cold water this injector wastes very 
little in starting. If the injector is hot or is filled with hot 
water when started, it will waste hot water till the injector is 
cooled by the water from the feed-supply, and will then work 



1 86 THERMODYNAMICS OF THE STEAM-ENGINE. 

as usual. If air leaks into the suction-pipe or if there is any- 
other interference with the normal action, the injector wastes 
water or steam till normal conditions are restored, when it 
starts automatically. 

The following table gives the results of tests made on an 
injector of this type:* 

TEST OF A SELLERS IMPROVED INJECTOR OF 1887, SIZE io£, 
LIFT 4 FEET, SUPPLY-WATER WEIGHED. 

MAXIMUM CAPACITY. 

Mean steam-pressure 30 60 122 151 200.5 

1. Gallons of water per hour 1912 2535 3517 3765 4005 

2. Temperature of supply-water . .. 67.0 67.0 54.0 50.0 50.5 

3. Temp, of delivered water 113.25 125.0 133.4 135.7 154.0 

4. Weight of water delivered) „„ _ , , 

H & , c * a\ 25.90 19.10 13.60 12.60 10.34 

per pound of steam used. . . ) J * * J J ^ 

MINIMUM CAPACITY. 

Mean steam-pressure 30 60 120 148 200.6 

5. Gallons of water per hour 765 937 1290 1432 1732 

6. Temperature of supply-water... 67.0 67.0 54.5 55.0 50.0 

7. Temp, of delivered water 171 212 238 250 263 

RANGE. 

Mean steam-pressure 30 60 121 149.5 200.5 

8. Per cent min. capacity of max.. 40.0 37.0 36.6 38.0 43.2 

9. Actual range in gals, perhr... 1147 1598 2227 2333 2273 
10. Per cent range of max. capacity. 60.0 63.0 63.3 62.0 56.8 

LIMITING TEMPERATURE OF WATER-SUPPLY. 

Mean steam-pressure 30 60 120 150 

11.. Limiting re-starting temp 130 135 122 120 

12. Limiting operating temp 139 144 137 133 

In these tests the amount of water supplied to the injector 
was measured; the amount of water delivered per pound of 
steam was calculated by aid of equation (220). 

Exhaust-steam Injectors. — Injectors supplied with ex- 
haust-steam from a non-condensing engine can be used to 
feed boilers up to a pressure of about So pounds. Above 
this pressure a supplemental jet of steam from the boiler must 



* 



The Railroad Gazette, Dec. 11, 1896. 



INJECTORS. 



I 8/ 



fXMM»S"\ V\tAW\ 



be used. Such an injector, as made by Schaffer and Buden- 
berg, is represented by Fig. 36; when 
used with low boiler-pressure this in- 
jector has a solid cone or spindle in- 
stead of the live-steam nozzle. To 
provide a very free overflow the com- 
bining-tube is divided, and one side is 
hung on a hinge and can open to give 
free exit to the overflow when the 
injector is started. When the injector 
is working it closes down into place. 
The calculation for an exhaust-steam 
injector shows that enough velocity 
may be imparted to the water in the 
delivery-tube to overcome a moderate 
boiler-pressure. FlG - 36. 

For example, an injector supplied with steam at atmos- 
pheric pressure, and raising the feed-water from 40 F. to 
100° F., will draw from the reservoir 




1 146.6 — 68.01 
68.01 — 8.06 



= 18.0 



pounds of water per pound of steam. Adiabatic expansion 
to the pressure of 7.8 pounds will reduce the steam to the 
quality 

642.7/965.8 \ 

9^9^ + °' 3127 " °' 2667 > = °' 9649 ' 



x n = 



The velocity of the steam will be 

V s = 52x32X778(1146.6— 0.9649X986.9— 150.6)^=1480 feet. 

Assuming the water to enter the combining-tube with a 
velocity of 30 feet, the velocity of the jet in the delivery-tube 
will be 

14 80+ 180 X 3 ~ t . 

V w — ; — 5 = 106 feet, 

1 -f- 18.0 



1 88 THERMODYNAMICS OF THE STEAM-ENGINE. 

and it can overcome a pressure of 



1 06 X 62.4 

= 75.0 pounds absolute. 



2 X 3 2 - 2 X 144 



Tests of Injectors. — The table opposite gives the results 
of tests made on several different injectors by Messrs. Bradlee 
and Blanchard,* in the laboratories of the Massachusetts 
Institute of Technology: 

Sizes of Orifices. 

Hancock: lifter, steam o. 114 

" forcer, steam 0.206 

" " water 0.149 

Lombard: steam 0.224 

" water 0.164 

Dodge: steam o. 161 

" water o. 131 

The experiments 15 to 21, inclusive, were made with im- 
proved methods of reducing the evaporation of the hot water 
delivered by the injector, and the results are more consistent 
and reliable than the preceding ones. It is apparent that the 
weight of steam used, which is obtained by taking the differ- 
ence between the weights of water supplied and delivered, is 
diminished by the evaporation, and that consequently the 
experimental quantity of water delivered by one pound of 
steam is made too large thereby: this explains part of the 
discrepancy of the first fourteen experiments. 

The calculation for Experiment 17 on the Dodge injec- 
tor, on the assumption that the pressure in the steam-orifice 
is 0.6 of the absolute boiler-pressure, and the steam in the 
supply-pipe contains three per cent of moisture, gives for the 
quality in the orifice 0.94, for the velocity of the steam-jet 

* Thesis, 1888. 



INJECTORS, 



I89 



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THERMODYNAMICS OF THE STEAM-ENGINE. 



1398 feet, and for the specific volume in the orifice 7.76 cubic 
feet. The discharge through the steam-nozzle by calculation 
is 91.7 pounds, while the experiment showed 93.1 pounds; 
which is fairly satisfactory, as the steam is determined by 
taking the difference between the water supplied to and that 
delivered by the injector. If the velocity of the water enter- 
ing the combining-tube is assumed to be 20 feet, then the 
velocity of the jet in the delivery-tube appears to be 144 feet, 
while the velocity required to overcome the delivery-pressure 
of 74 pounds by the gauge or 88.7 pounds absolute is 115 
feet. But if the jet is assumed to be clear water with a 
density of 62.4 pounds, the velocity calculated from the area 
of the steam-orifice and the quantity of water delivered will 
be 78 feet per second; consequently the density of the jet 
will be about 0.60 that of water. 

Water-injector. — Fig. 37 represents a device called a 
water-injector, in which a small stream of water in the pipe 




Fig. 37. 



M flowing from the reservoir R raises water from the reservoir 
R" to the reservoir R '. 

Let one pound of water from the reservoir R draw y 
pounds from R" , and deliver 1 -f- y pounds to R' . Let the 
velocity of the water issuing from A be v\ that of the water 



INJECTORS. I9I 

entering from R" be v^ at N\ and that of the water in the 
pipe O be v x . The equality of momenta gives 

v+yv 9 = (i+y)v t (225) 

Let x be the excess of pressure at M above that at N 
expressed in feet of water; then 

V* = 2gX\ (226) 

v* = 2g(H+x); (227) 

v i — 2 g^ h + *) (228) 

Substituting in equation (225), 



VH-\-x+yVx = (1 +y)V k + x; 



VH + x — Vk + x 
y k -f- x — V x 

It is evident from inspection of the equation (229) that y 
may be increased by increasing x, for example, by placing 
the injector above the level of the reservoir so that there may 
be a vacuum in front of the orifice A. 

If the weight G of water is to be lifted per second, then 

Q 

— pounds per second must pass the orifice A, G pounds the 

space at N, and (1 -\- -JG pounds through the section at 0\ 

which, with the several velocities v, z\, and v lt give the data 
for the calculation of the required areas. 

PROBLEM. — Required the calculation for a water-injector 
to raise 1200 gallons of water an hour, H = 96 ft., h = 12 ft., 

x = 4 ft. 



Vx = V^ = 2\ VH X x — Vioo = 10; V/i + x= Vi6 = 4; 



10 — 4 

4 — 2 



192 THERMODYNAMICS OF THE STEAM-ENGINE. 

The velocities are 



v = V2 X 32.2 X 100 = 80.25 feet per second ; 
v x = V2 X 32.2 X 16 = 32.10 feet per second ; 
v^ = V2 X 32.2 X 4 = 16.05 feet per second. 

1200 gallons an hour — 0.04452 cubic feet per second. 
The areas are 



0.04452 
a = — - — - = 0.000185 square feet ; 

3 X 80.25 D H 

4 X 0.04452 _ 
= 0.00185 square feet; 



3 X 32.10 
0.04452 



= 0.00277 square feet. 



16.05 

The diameters corresponding to the velocities v and v 1 are 

d = 0.18 of an inch ; 
d l = 0.58 of an inch. 

The area # 2 is of annular form, having the area 0.4 of a 
square inch. 

Ejector. — When the injector is used for raising water 
where there is no advantage in heating the water, it is a very 
wasteful instrument. The efficiency is much improved by 

arranging the instrument as in 
Fig. 38, so that the steam-nozzle 

t A shall deliver a small stream of 

water at a high velocity, which, 
as in the water-injector, delivers 
a larger stream at a less velocity. Each additional conical 
nozzle increases the quantity at the expense of the velocity, 
so that a large quantity of water may be lifted a small height. 
Ejectors are commonly fitted in steamships as auxiliary 
pumps in case of leakage, a service for which they are well 




INJECTORS. 193 

fitted, since they are compact, cheap, and powerful, and are 
used only in emergency, when economy is of small conse- 
quence. 

Ejector-condensers. — When there is a good supply of 
cold condensing water, an exhaust-steam injector, using all 
the steam from the engine, may be arranged to take the place 
of the air-pump of a jet-condensing engine. The energy of 
the exhaust-steam flowing from the cylinder of the engine to 
the combining-tube, where the absolute pressure is less and 
where the steam is condensed, is sufficient to eject the water 
and the air mingled with it against the pressure of the atmos- 
phere, and thus to maintain the vacuum. 

For example, if the absolute pressure in the exhaust-pipe 
is 4 pounds, and if the temperatures of the injector and the 
delivery are 6o° F. and ioo° F., then the velocity of the 
steam-jet will be 1226 feet, and each pound of steam will 
draw into the injector 24 pounds of water. If the injection 
water enters with a velocity of 20 feet, the velocity of the 
water-jet will be 68.2 feet per second, which can overcome 
the pressure of 31 pounds absolute. 



CHAPTER X. 
HOT-AIR ENGINES AND GAS-ENGINES. 

Engines of Maximum Efficiency. — In order to have the 
maximum efficiency, an engine must work on such a cycle 
that its working substance shall always have the temperature 
of the source of heat when acquiring heat, and the tempera- 
ture of the refrigerator when rejecting heat; that is, the 
engine must be reversible. 

The older forms of hot-air engines all had the source of 
heat at one constant temperature and the refrigerator at 
another lower constant temperature. To have the maximum 
efficiency it was required that the working substance should 
receive heat from external sources at one temperature, and 
reject heat to external sources at one temperature only. 

Carnot's engine is the only simple engine which can fulfil 
these conditions when air is the working substance. The 
cycle of that engine has never been adopted in practice, since 
it involves incompatible requirements; that is, the isothermal 
changes should be very slow and the adiabatic changes should 
be very rapid, to make the cycle of an actual engine approxi- 
mate to the ideal cycle. And, further, Carnot's cycle for a 
perfect gas is so attenuated from the fact that the angle 
between an isothermal and an adiabatic where they cross is 
very acute, that the mean effective pressure is small, and the 
cylinder of the engine to work on such a cycle would be very 
large for the power developed. 

By aid of a device called a regenerator or economizer, 
actual engines have been made which have an ideal cycle of 
maximum efficiency. Such a cycle is represented by Fig. 39. 

194 



HOT-AIR ENGINES AND GAS-ENGINES. 



195 




The curves DC and AB are isothermals, which form those 
parts of the cycle during which heat is re- p 
ceived from the source and rejected to the 
refrigerator. The curves BC and DA cor- 
respond to the adiabatic lines of Carnot's 
cycle, and must fulfil the condition that the 
heat given to the regenerator during one 
operation, as that represented by BC, must FlG - 39- 

be equal to the heat received from the regenerator during the 
converse operation DA. 

To obtain the relation between the curves BC and DA, 
draw the intermediate isothermals XZ and WY with a differ- 
ence of temperature dt. The heat received by one unit of 
weight of the working substance in passing from W to X is 

dQ = c p dt + mdp, (230) 

and that rejected from Z to Y is 

dQ = c p dt -f- *# 'dp' (231) 

The required condition will be fulfilled by making equa- 
tion (230) equal to equation (231), so that 

mdp = m'dp' . 

Substituting for m from equation (53) gives 

(dv\ 



©*-( 



dV '\ M 



Deducing the values of the partial differential coefficients 
from the characteristic equation for a gas, and substituting, 
we have 

R R , 

-dp = j4P ; 



dp 

7 

P 



df_ 

/ ; 

Cp\ 



(232) 



196 



THERMOD YNAM1CS OF THE STEAM-ENGINE. 




That is, the required relation is that the ratio of the pressures 
at the points cut by any isothermal from the paths DA and 
BC must be constant. 

Stirling's Engine. — This engine was invented in 18 16, 
and was used with good economy for a few years, and then 
rejected because the heaters, which took the place of the 
boiler of a steam-engine, burned out rapidly. It is described 
and its performance given in detail by Ran- 
kine in his Steam-engine. An ideal sketch is 
_ given by Fig. 40. E is a displacer piston 
filled with non-conducting material, and work- 
ing freely in an inner cylinder. Between 
this cylinder and an outer one from A to C 
is placed a regenerator made of plates of 
metal, wire screens, or other material, so 
arranged that it will readily take heat frcm 
Fig. 40. or yj e id heat to air passing through it. At 

the lower end both cylinders have a hemispherical head; that 
of the outer cylinder is exposed to the fire of the furnace, and 
that of the inner is pierced with holes through which the air 
streams when displaced by the plunger. At the upper end 
there is a coil of pipe through which cold water flows. The 
working cylinder H has free communication with the upper 
end of the displacer cylinder, and consequently it can be oiled 
and the piston may be packed in the usual manner, since only 
cool air enters it. 

In the actual engine the cylinder H is double-acting, and 
there are two displacer cylinders, one for each end of the 
working cylinder. 

If we neglect the action of the air in the clearance of the 
cylinder H and the communicating pipe, we have the follow- 
ing ideal cycle. Suppose the working piston to be at the 
beginning of the forward stroke, and the displacer piston at 
the bottom of its cylinder, so that we may assume that the 
air is all in the upper part of that cylinder or in the refriger- 
ator, and at the lowest temperature T^ the condition of one 




HOT- A IE ENGINES AND GAS-ENGINES. ig7 

pound of air being represented by the point D of Fig. 39. 
The displacer piston is then moved quickly by a cam to the 
upper end of the stroke; while the working p 
piston moves so little that it may be con- 
sidered to be at rest. The air is thus all 
driven from the upper end of the displacer 
cylinder through the regenerator, from 
which it takes up heat abandoned during 
the preceding return stroke, thereby acquir- FlG - 4*. 

ing the temperature T x , and enters the lower end of that 
cylinder. During this process the line AD of constant 
volume is described on Fig. 41. When this process is com- 
plete, the working cylinder makes the forward stroke, and the 
air expands at constant temperature, this part of the cycle 
being represented by the isothermal AB of Fig. 41. At the 
end of the forward stroke the displacer piston is quickly 
moved down, thereby driving the air through the regenerator, 
during which process heat is given up by the air, into the 
upper part of the displacer cylinder; this is accompanied by 
a cooling at constant volume, represented by the line BC. 
The working piston then makes the return stroke, compress- 
ing the air at constant temperature, as represented by the 
isothermal line CD, and completing the cycle. 

To construct the diagram drawn by an indicator, we may 
assume that in the clearance of the cylinder H, the communi- 
cating pipe, and refrigerator there is a volume of air which 
flows back and forth and changes pressure, but remains at the 
temperature 7~ 2 . If we choose, we may also make allowance 
for a similar volume which remains in the waste spaces at the 
lower end of the displacer cylinder, at a constant tempera- 
ture T x , 

In Fig. 42, let ABCD represent the cycle of operations, 
without any allowance for clearance or waste spaces; the 
minimum volume will be that displaced by the displacer 
piston, while the maximum volume is larger by the volume 
displaced by the working piston. Let the point E represent 



198 



THERMODYNAMICS OF THE STEAM-ENGINE. 



the maximum pressure, the same as that at A ; and the united 
volumes of the clearance at one end of the working cylinder, 
of the communicating pipe, of the clearance at the top and 

bottom of the displacer cylinder, and 
the volume in the refrigerator and 
regenerator. Each part of this com- 
bined volume will have a constant 
temperature, so that the volume at 
different pressures will be represented 
c ' v - by the hyperbola EF. To find the 
FlG ' 42, actual diagram A' B' C D' , draw any 

horizontal line, as sy, cutting the true diagram at u and v, 
and the hyperbola EF at /; make ux and vy equal to st\ then 
x and y are points of the actual diagram. The indicator will 
draw an oval similar to A'B'C'D' with the corners rounded. 

To show that the diagram, Fig. 41, fulfils the condition 
for maximum efficiency, drawn an intermediate isothermal 
XY. Since DA and BC are lines of constant volume, 










Tl-Py. 

T c ~Pc' 



P* — pPy - CPr 



The diagram in Fig. 43 was reduced from an indicator- 
diagram from a recent hot-air engine made on the same prin- 
ciple as Stirling's hot-air engine. To avoid destruction of the 
lubricant in the working cylinder 
Stirling found it advisable to con- 
nect only the cool end of the dis- 
placer cylinder with the working 
cylinder, and had two displacer 
cylinders for one working cylinder. 
It has been found that a good mineral oil can be used to 
lubricate the displacer piston, and that the hot end also of the 
displacer cylinder can be advantageously connected with the 



Fig. 43. 



HOT-AIR ENGINES AND GAS-ENGINES. 1 99 

working cylinders, of which there are two. Thus each work- 
ing cylinder is connected with the hot end of one displacer 
cylinder and with the cool end of the other displacer cylinder. 

The distortion of the diagram Fig. 43 is due in part to the 
large clearance and waste space, and partly to the fact that 
the displacer pistons are moved by a crank at about jo° with 
the working crank. 

A test on the engine mentioned by Messrs. Underhill 
and Johnson * showed a consumption of 1.66 of a pound of 
anthracite coal per horse-power per hour; but the friction of 
the engine is large, so that the consumption per brake horse- 
power is 2.37 pounds. This engine, like the original Stirling 
engine appears to have given much difficulty from the burn- 
ing of the heaters. The difficulty is likely to be more serious 
with large than with small engines, as the volume of the dis- 
placer cylinders increases more rapidly than the heating 
surface. 

Ericsson's Engine. — This engine consists essentially of a 
working cylinder, a compressing-pump, and a reservoir. To 
give a perfect ideal cycle, a regenerator and a refrigerator are 
required. The pump, which must have a water-jacket which 
acts as a refrigerator, draws air from the atmosphere at con- 
stant pressure, compresses it at constant 
temperature, and forces it into the reser- | F a b 
voir under constant pressure. The pump 
cycle is represented by the diagram EDAF 
(Fig. 44). The engine draws air from 
the reservoir through the regenerator, 
during which process it is heated from IG * 44# 

the temperature T 7 to T t ; the supply is then cut off by a 
slide-valve, and the air in the cylinder expands at constant 
temperature down to the atmospheric pressure. On the 
return stroke the air is forced from the cylinder at constant 
pressure through the regenerator, being thereby cooled to the 

* Thesis, 1889. 



\ 


\ 


\g 


^C 


E 


D 


H 











v' 



200 THERMODYNAMICS OF THE STEAM-ENGINE. 

temperature T t . The engine cycle is represented by the 
diagram FBCE. The diagram of effective work is ABCD, 
which fulfils the condition of maximum efficiency, since AD 
and BC are isothermals, and AB and CD are lines of constant 
pressure. 

The actual engine does not expand down to the atmos- 
pheric pressure, so that the diagram is cut short by a line like 
GH. Also, the clearances of the two cylinders introduce 
irregularities and modifications of the diagram. 

An engine of 300 horse-power designed by Ericsson for a 
ship named after himself used about 2 pounds of coal per 
indicated horse-power per hour; but its friction was exces- 
sive, and it was finally replaced by a steam-engine. Even 
when this type of engine is provided with a regenerator and 
is made to work on a closed cycle with two reservoirs, one 
at a high pressure from which air is supplied to the working 
cylinder, and one at a lower pressure from which the pump is 
supplied, it may be expected to give a poorer economy than 
an engine working on the Stirling cycle. 

Gas-engines. — The chief difficulty with hot-air engines is 
to transmit heat to and from the working substance. In gas- 
engines this difficulty is removed by mixing the fuel with the 
air (so that heat is developed in the working substance itself), 
and by rejecting the hot gases after they have done their 
work. The fuel may be illuminating-gas, fuel-gas, or vapor 
of a volatile liquid like gasoline. It will be shown that the 
specific volume and the specific heat of the mixture of air and 
gas, both before and after the heat is developed by combus- 
tion, are not very different from the same properties of air. 
The general theory of gas-engines may therefore be devel- 
oped on the assumption that the working substance is air, 
which is heated and cooled in such a manner as to produce 
the ideal cycles to be discussed, as is done by Clerk. * 

Experience has shown that in order to work efficiently, 
the mixture of gas and air supplied to a gas-engine must be 

* Gas-wgines, Dugald Clerk. 



HOT-AIR ENGINES AND GAS-ENGINES. 201 

compressed to a considerable pressure before it is ignited. 
This may be done either by a separate compressor or in the 
cylinder of the engine itself; the second type of engines, of 
which the Otto engine is an example, is the only successful 
type at the present time; the other type has some advantages 
which may lead to its development. 

Gas-engine with Separate Compressor. — This engine 
has a compressor, a reservoir, and a working cylinder. When 
run as a gas-engine a mixture of gas and air is drawn into a 
pump or compressor, compressed to several atmospheres, and 
forced into a receiver. On the way from the receiver to the 
working cylinder the mixture is ignited and burned so that 
the temperature and volume are much increased. After 
expansion in the working cylinder the spent gases are ex- 
hausted at atmospheric pressure. 

The ideal diagram, represented by Fig. 45, has a general 
resemblance to that of Ericsson's engine, but it differs essen- 
tially because the expansion and compression curves are 
adiabatics instead of isothermal lines. ED represents the 
supply of the combustible mixture to the 
compressor, DA is the adiabatic compres- 
sion, and AF represents the forcing into 
the receiver. FB represents the supply 
of burning gas to the working cylinder, 
BC represents the expansion and CE the 
exhaust. In practice this type of engine l ' 

always has a release, represented by GH, before the expan- 
sion has reduced the pressure of the working substance to 
that of the atmosphere. 

This type of engine has been used as an oil-engine by 
supplying the fuel in the form of a film of oil to the air after 
it has been compressed. In such case the compressor draws 
in air only, and there is not an explosive mixture in the 
receiver. The Brayton engine when run in this way could 
burn crude petroleum or, after it was started, could burn 
refined kerosene. Its chief defect appears to have been in- 



p 

F A 


B 


\G 




V 




^C 


E D 





H 


v r 



202 THERMOD YNAMICS OF THE STEAM-ENGINE. 

complete combustion and consequent fouling of the cylinder 
with carbon. 

The effective cycle may be considered to be represented 
by the diagram ABCD (Fig. 45), and may be assumed to be 
produced in one cylinder by heating the air from A to B, by 
cooling it from C to D, and by the adiabatic expansion and 
compression from B to C and from D to A. If T a and T b are 
the absolute temperatures corresponding to the points A and 
B, then the heat added from A to B is 

C p\lb — J- a) J 

and the heat withdrawn from C to D is 

c p {T c - T d ), 
so that the efficiency of the ideal cycle is 

V ~~ c p {T h -T a ) - 1 T b -T a - l233 > 

But since the expansion and compression are adiabatic, 

T b ~\pJ ' T a -\pJ 
but p c = p d and p b = p a , therefore 



K-I 



T b ~ T a dna T b -T a ~ T. ~ W 
so that the equation for efficiency becomes 



*=!-£)■ •• (234> 



HOT-AIR ENGINES AND GAS-ENGINES. 203 

For example, with the pressure in the reservoir at 90 
pounds above the atmosphere the efficiency is 

1.405- 1 
14.7 \ *** 



Tf = I — I — = O.43. 

' \ .I4.7 +9O/ ^ J 

When the cycle is incomplete the expression for the 
efficiency is not so simple, for it is necessary to assume cool- 
ing at constant volume from G to H (Fig. 45), and cooling at 
constant pressure from H to D\ so that the heat rejected is 

c v { T g — T h ) + c p { T k — T d ), 

and the efficiency becomes 

l -(T s - T h ) + (7; -T d ) 

V=I T j=r . . . . (235) 

1 b — J- a 

For example, let it be assumed that the pressure at A is 
90 pounds above the atmosphere, that the temperature at B 
is 2500 F., and that the volume at G is three times the 
volume at B. 

First, the temperature at A is 

K— I 0.40$ 

provided that the temperature of the atmosphere is 6o° F. 
The temperature at G is 

T £ = TJ^J' 1 = 2 96o. 7 (i) ' 405 = 1897, 

and the pressure at G is 

(v\ K (1 V 405 

P S = P\-) =047 + 9O)bJ =22.4 pounds, 



204 



THERMOD YNAMICS OF THE STEAM-ENGINE. 



so that the temperature at H is 
and finally the efficiency is 



s p g y/ 22.4 



1247, 



77=1- 



1.405 



(1897 - 1247) + 1247 - 520.7 



0.42. 



2960.7 — 917 

Gas-engine with Compression in Cylinder. — All succes- 
ful gas-engines of the present day compress the explosive 
mixture of gas and air in the working-cylinder, and as they 
take gas at one end of the cylinder only they must make four 
strokes for each explosion. The first forward stroke of the 
piston from the head of the cylinder draws in the mixture of 
gas and air, which is compressed on the return stroke; at the 
completion of this return stroke the mixture is ignited and 
the pressure rises very rapidly; the next forward stroke is the 
working stroke, which is succeeded by an exhaust stroke to 
expel the spent gases. In almost all engines these four 
strokes are of equal length, for the advantage of making them 
of unequal length, as required for the best ideal cycle, is more 
than counterbalanced by the mechanical difficulty of produc- 
ing unequal strokes. 

The most perfect ideal cycle, represented by Fig. 46, has 

four strokes of unequal length so 
arranged that the piston starts from 
the head of the cylinder when gas is 
drawn in, and the pressure in the 
cylinder is reduced to that of the 
atmosphere before the exhaust stroke. 
Thus there is the filling stroke, rep- 
resented by EC; the compression 
stroke, represented by CD; the work- 
ing stroke, represented byAB; and the 
exhaust stroke, represented by BE. 
The effective cycle is ABCD, which may be considered to 




HOT-AIR ENGINES AND GAS-ENGINES. 20 5 

be performed by adding heat at constant volume from D to 
A, and withdrawing heat at constant pressure from B to C, 
together with the adiabatic expansion and compression AB 
and CD. 

The heat added under this assumption is 

c v ( T a — T d ), 
and the heat rejected is 

c p (T b — T c ), 
so that the efficiency is 

dT.- T d )-c p {T t - T J _ T t - T c 

V c,{T a -T d ) T a -T d - (230) 

If the temperature at A and the pressure at D are assumed, 
then it is necessary to make preliminary calculations of the 
temperatures at D and at B before using equation (236) by 
the equations 

■ K — I 

T 4 =rl*A'~ r -> (237) 

w 

A=Ap; (238) 

I d 



K —\T 



Pa 



T b =T a (£t\ K (239) 



For example, if the pressure at the end of compression is 
90 pounds above the atmosphere, and the temperature at the 
end of the explosion is 2500 F., then 



0.405 



T d = (6o + 4 6o. 7 )(l±L±22)^ = 9 i 7 , 

provided that the temperature of the atmosphere is 6o° F. 

2500+460.7 , 

p a = 104.7-? — ^y — - = 338 pounds ; 



206 



THERMODYNAMICS OF I HE STEAM-ENGINE. 



0.405 



7- d = ( 25 oo + 46o.7)(!g)'- 405 =ii99; 

I IOQ — 520.7 

n = 1 — 1.405 — ^ — ' = 0.55. 

2960.7 — 917 J 

If the expansion is not carried to the 
atmospheric pressure, then the diagram 
shows a release at the end of the stroke, 
as in Fig. 47, and the cycle must be con- 
sidered to be formed by adding heat as 
before at constant volume, but by with- 
drawing heat at constant volume to cause 
a loss of pressure from B to G, and by 
withdrawing heat at constant pressure 
during the process represented by GC. 
The heat rejected becomes, therefore, 

c v ( T b — T g ) + c,( T g - T c ), 

and the efficiency is 

_ c v (T a - T d ) - c v {T b - T g ) - c P {T g - T c ) 







T h 



c v (T a — T d ) 
T g +K{T g -T c ) 



T a - T d ) 



(240) 



Assuming, as before, the pressure at D and the tempera- 
ture at A, it becomes necessary to find the temperatures at B 
and at G as well as the temperature at D\ this last may of 
course be found by equation (237). If the pressure at B is 
assumed also, then equations (238) and (239) may be used as 
before to find T b , and T g may be found by the equation 



T,= T b 



(241) 



For example, let it be assumed that the expansion ceases 
when the pressure becomes 20 pounds above the atmosphere, 



HOT-AIR ENGINES AND GAS-ENGINES. 



207 



the other conditions being as in the previous example. 
Then 

0405 
T, = (2500 + 4 6o. 7 )(^-± i - V' 405 = 1 536 ; 



^=1536^1 = 650; 



and 



V = 1 



1536-650+ 1.405(650 - 520.7) 
2960.7 — 917 



= O.48. 



Though not essential to the solution of the example, it is 
interesting to know that the volume at C is 



9 °+i47 V 
14.7 



= 4 + 



times the volume at D, and that the volume at B is 



v 34-7' 

times the volume at A. 

When, as in common practice, the four 
strokes of the piston are of equal length 
the diagram takes the form shown by Fig. 
48; the effective cycle may be considered 
to be equivalent to heating at constant 
volume from D to A and cooling at constant 
volume from B to C, together with adiabatic 
expansion and compression from A to B and 
from C to D, 

The heat applied is 

c v ( T a — T d ), 
and the heat rejected is 

c v (T b — T c \ 




Fig. 48. 



208 THERMODYNAMICS OF THE STEAM-ENGINE. 

so that the efficiency is 

_ c v\ T* — T d ) — c v { T b — T c ) T h — T c , % 

v ciT a - Ti ) - 1 ~t^t; (24I) 

Since the expansion and compression are adiabatic, we 
have by equation (79) 

T b vf ~ x = T a v a < - * , and 7> c « ~ * = 7>/ - 1 ; 

but the volumes at A and D are equal, as are also the volumes 
at B and C; consequently by division 

T.~2 t ' 

consequently 

T h -T c T b T c (vA—* 



T a — T d T a T d \v L 
and the expression for efficiency becomes 



> 



7 = 1- (J) (242) 



V C 



which shows that the efficiency depends only on the compres- 
sion before explosion. 

For example, if the volume of the clearance or compres- 
sion space is one-third of the piston displacement, so that v d 
is one-fourth of v then the efficiency is 

/i\ 0.405 

" =I -( 4 ) =° 43 - 

The pressure at the end of compression is 
A =4g)"= i^)" 405 = 103-1 




HOT-AIR ENGINES AND GAS-ENGINES. 209 

pounds absolute, or 88.4 pounds by the gauge. The calcu- 
lated efficiency is therefore not much less than the efficiencies 
found for other examples; it is notable that the efficiency is 
nearly the same as that calculated on page 205 for an engine 
with separate compression to 90 pounds by the gauge. For 
the case in hand, however, the pressure after explosion, which 
depends on the temperature, may exceed 300, as was shown 
in the preceding examples, for engines with compression in 
the cylinder. 

The diagrams from engines of this type resemble Fig. 49, 
which was taken from an Otto engine tested under the direc- 
tion of Professor Thurston. Dur- 
ing the filling stroke the pressure is 
gradually reduced below that of 
the atmosphere; the explosion is 
nearly but not quite instantaneous, 
as is shown by the explosion line FlG - 49 - 

leaning toward the right and by the rounded curve joining it 
to the expansion line; there is a release before the end of the 
stroke and a little back-pressure at the beginning of exhaust. 
But the greatest difference between this diagram and the ideal 
diagram is found in the expansion curve, which is far from 
being an adiabatic, for the combustion does not cease when 
the maximum pressure is attained, but continues throughout 
the stroke; and at the same time heat is taken up energeti- 
cally by the walls of the cylinder, which are cooled by a water- 
jacket to avoid overheating. These two effects, after-burning 
and loss of heat to the water-jacket, may nearly compensate, 
and then the expansion curve will resemble the adiabatic line 
in appearance. 

The engine from which this diagram was taken had a 
diameter of 8.5 inches and a stroke of 14 inches. At 158 
revolutions per minute it indicated 9.6 horse-power and 
developed 8.1 horse-power on a brake. It consumed 24.5 
cubic feet of illuminating-gas per horse-power per hour. 



2IO 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Professor Thurston estimates the distribution of heat as fol' 

lows: 

Work indicated in cylinder ........ 17.0 

Heat lost to cylinder walls.. , 52.0 

Heat carried away by exhaust gases 15.5 

Heat lost by conduction and radiation 15.5 



1 00.0 



so that the actual efficiency is 17 per cent. 

The compression space was 38 per cent of the total 
cylinder volume, so that the efficiency of the ideal cycle was 
42 per cent, and the ratio of the efficiencies was about 0.40. 
More modern and especially larger engines use less gas per 
horse-power per hour, and show a better efficiency both abso- 
lutely and relatively. 

There are two questions that should now receive consid- 
eration if our calculations of ideal efficiency are to be taken 
as a guide either in the choice of a type of engine or the pro- 
portions of a type. They are the composition of the mixture 
in the cylinder both before and after explosion, and the prob- 
able temperature after explosion. As for the composition of 
the mixture, Clerk gives the following table showing the com- 
position of Manchester illuminating-gas, the oxygen required 
for combustion, and the volume of the products after combus- 
tion, all reduced to normal pressure and temperature. 
ANALYSIS OF MANCHESTER COAL-GAS. (Bunsen and Roscoe.) 







Amount required 
for Complete 
Combustion. 


Products. 




vols. 

45-58 

34-9 
6.64 
4.08 
2.38 
0.29 
2.46 
3.67 


vols. O. 
22.79 
69.8 

3.32 
12.24 
14.28 

0.43 


vols. 

45.58, H 2 
104.7, C0 2 & H 2 

6.64, C0 2 
16.32, C0 2 & H a O 
19.04, C0 2 & H 2 

0.58, H 2 0& S0 2 

2.46 

3.67 


Marsh-gas CH4 


Carbonic oxide CO 




Sulphuretted hydrogen, H 2 S. 










Total 


100.00 


122.86 O 


i98.99,C0 2 ,H 2 0&SO a 





HOT AIR ENGINES AND GAS-ENGINES. 211 

It appears in this case that the volume of the gas and 
oxygen is reduced by combustion from 223 to 199, both being 
at freezing-point, so that the error of neglecting such a change 
would appear to be about 10 per cent. But the gas is mixed 
with from 7 to 12 volumes of air when it is introduced into 
the cylinder of the engine so that the error is probably less 
than two per cent in any case; though of course it depends 
on the gas used, which may be illuminating-gas distilled from 
coal, so-called water-gas, or crude gas made in a special gas- 
producer. 

To find the most economical mixture of gas and air and 
the probable temperature after combustion, Clerk made a 
large number of experiments on illuminating-gas distilled 
from coal. He found the pressure immediately after explo- 
sion to vary from 40 to 90 pounds per square inch above the 
atmosphere, depending on the kind of gas and the proportions 
of the mixture. Since this change of pressure takes place at 
constant volume (neglecting change of volume due to combus- 
tion) the absolute temperatures must be proportional to the 
absolute pressures, giving, for a temperature of 62 F. before 
explosion, 

(460.7 + 6 2 )i4rZ±i2^ I940( 

(460.7 + 62)M^_±^= 3720, 

for the absolute temperatures, or from 1500 F. to 3000 F. 

The temperature after explosion appears to be largely 
controlled by the phenomenon of dissociation, for a calculation 
of the temperature of combustion on the assumption that the 
total heat of combustion of the gas is developed during the 
explosion and is used in raising the temperature of the 
products of combustion gives temperatures which are about 
double those just stated. When the temperature of the gases 
after explosion has been reduced by expansion and by con- 



212 



THERMO D YNAMICS OF THE STEAM-ENGINE. 



duction and radiation to the cylinder of the engine, the com- 
bustion may be continued and may extend throughout the 
stroke; this phenomenon is called after-burning. 

The performance of gas-engines has been improved by 
(i) raising the compression before explosion; (2) increasing 
the size of the engine; (3) clearing the cylinder of spent 
gases before introducing a new charge of gas. 

The following table of tests on Crossley-Otto engines of 
about the same size illustrates the advantage of increasing the 
compression: 



Diameter of cylinder 

Stroke 

Ratio of compression space to piston dis- 
placement 

Pressures of compression above atmosphere. 

Gas per horse-power per hour 

Actual efficiency 

Theoretical efficiency 

Ratio of efficiencies 



9.0 


9.5 


18.0 


18.0 


0.6 


0.4 


38.0 


61.6 


24.0 


20.5 


0.17 


0.21 


0.33 


0.40 


0.51 


0.53 



7.0 

150 

0.34 

87-5 

14.8 
0.25 
0.428 
0.58 



Since the greatest waste of the gas-engine is due to the 
heat carried away by the water-jacket, it is to be expected 
that a large engine will show a better performance than a 
small engine, since the area of the water-jacket for such an 
engine will have a smaller ratio to the volume of the cylinder. 
This is illustrated by the following results from engines of 
different sizes, but having about the same degree of compres- 
sion: 



Diameter of cylinder 

Stroke 

Relative capacity 

Actual efficiency 

Theoretical efficiency 
Ratio of efficiencies . . 



The amount of heat added to the gases during explosion 
depends to a large extent on the temperature before the 
explosion, since the temperature after explosion is largely 
controlled by dissociation. It is therefore important that the 
engine shall be charged with a cool mixture of air and gas. 




HOT-AIR ENGINES AND GAS-ENGINES. 213 

But the compression, or cartridge space, as it is sometimes 
called, is commonly rilled with hot spent gases which mingle 
with and raise the temperature of the fresh charge. Three 
methods of meeting this difficulty have been tried. 

1. In the Clerk engine the charge of gas and air is drawn 
into an auxiliary cylinder by a piston driven by a crank which 
is ninety degrees ahead of the motor-crank. The charge is 
lightly compressed and is forced into the motor-cylinder just 
before the motor-piston reaches the end of the stroke, and 
drives out the spent gases through holes in the walls of the 
cylinder which are overrun and uncovered by the piston at 
the proper time. The return stroke of the motor-piston 
compresses the charge, which is exploded at the beginning of 
the next forward stroke. This engine therefore makes an 
explosion for every revolution. In practice it is found that 
the fresh charge tends to mingle with the spent gases and that 
unburned gas escapes in the exhaust, so that there is little if 
any gain in efficiency over the Otto engine. 

2. The Atkinson engine has a double-beat linkage inter- 
posed between the piston and the crank, so that the piston 
makes four strokes of unequal length for each revolution. The 
diagram from the Atkinson engine has Fig. 47 for its ideal 
type. In the first place the piston has a very small clearance 
at the end of the exhaust stroke and the beginning of the 
filling stroke; and also the expansion reduces the pressure 
down towards that of the atmosphere before release. It 
appears that this engine showed marked improvement over 
the Otto engines with which it was compared, but the com- 
plicated linkage required to make the four unequal strokes 
gave much trouble, especially for large engines. More recent 
engines of the Otto type have, by increased size and compres- 
sion, given better economy than the Atkinson engines which 
were made and tested. A comparison of the examples on 
pages 207 and 208 shows that the Otto cycle with sufficient 
compression may be expected to give nearly as good results 
as an engine with more expansion. 



214 THERMODYNAMICS OF THE STEAM-ENGINE. 

3. The third attempt to clear the cartridge space is found 
in the Crossley scavenging-engine, in which, advantage is 
taken of the fact that the exhausts from the cylinder of an 
Otto engine set up a pulsation in the exhaust-pipe, and that 
between successive exhausts the pressure in that pipe is less 
than the pressure of the atmosphere. By making the ex- 
haust-pipe of the right length (about 65 feet) the time of the 
reduced pressure between the pulsations may be made to 
coincide with the completion of the exhaust stroke. If the 
inlet air-valve is opened at this time fresh air sweeps through 
the cylinder and clears it of spent gases before the next 
charge of gas and air is drawn in. Clerk considers that a gain 
of five per cent may be attained by properly clearing the 
cylinder of spent gases ; when larger gains have been reported 
they have been due in part to increased compression or in- 
creased size. 

Gas-producers. — Illuminating gas must be refined so as 
to remove impurities which would give an offensive odor when 
burning, and is consequently an expensive fuel, and can 
seldom compete on even terms with coal burned to make 
steam. Large gas-engines have been supplied with a cheap 
gas made specially for them by distillation and partial combus- 
tion of coal and coke. The Dawson gas-producer, which has 
been successfully used, in practice has a closed furnace into 
which anthracite or coke is charged through a hopper and 
charging-valve. Air is forced into the furnace under a slight 
pressure by a steam-jet from a special small steam-boiler. 
The fire is kept about 18 inches thick to give incomplete 
combustion, and gives off a gas which in a certain instance had 
the composition: 

Dowson Gas. Ideal Gas. 

Nitrogen 48.98 45.0 

Carbon monoxide 25.07 39.0 

Hydrogen 18.73 16.0 

Marsh-gas 0.31 

Olefiant gas 0.31 

Carbon dioxide 6.57 

Oxygen 0.03 

* 

100.00 



HOT-AIR ENGINES AND GAS-ENGINES. 21 5 

The ideal gas which would be produced if such a producer 
worked perfectly with pure carbon is given for the sake of 
comparison. A French analysis of Dowson gas gave over n 
per cent of carbon dioxide. 

After the gas is formed it is passed through a scrubber of 
coke sprayed with water to cool it and remove the large pro- 
portion of sulphur, which would eventually injure the cylinder. 
From the scrubber the gas passes to a gas-holder, and from 
the holder through another scrubber to the engine. This 
gas-producer cannot be run with coal which produces a tarry 
distillate, and has consequently been restricted to anthracite 
and coke. Other producers have been devised which are 
intended, by mingling the distillate from the coal with steam 
and passing it to incandescent fuel, to break it up into lighter 
compounds like marsh-gas and olefiant gas. Such producers 
if successful could use bituminous and other cheaper coals. 

The Dowson gas, from its composition, requires somewhat 
more than its own volume of air for complete combustion, and 
in general will give somewhat less mean effective pressure 
than illuminating-gas. Thus a scavenging-engine with Dow- 
son gas gave 97.4 pounds mean effective pressure, while with 
coal-gas it gave 1 1 3. 5 pounds. This was a large engine with 
diameter of 17 inches and a stroke of 24 inches; smaller en- 
gines may be expected to give about 65 pounds effective pres- 
sure with Dowson gas. 

A recent test by Professor Spangler of a 100-H.P. Otto 

engine using gas made in a Taylor producer showed that the 

producer developed 69 per cent of the heat of the coal, and 

that the engine used 1.3 pounds of coal per horse-power per 

hour when developing 130 horse-power. The coal had the 

following composition: 

Moisture 4.20 

Volatile matter 6.88 

Fixed carbon 80.41 

Ash 8.51 

Sulphur 0.74 

100.74 



2l6 THERMODYNAMICS OF THE STEAM-ENGINE. 

A test by Mr. Dowson of a Crossley engine showed one 
pound of coal per horse-power per hour; but as the coal in the 
producer was estimated at the beginning and end of the test, 
and as the test was only eight hours long, this result cannot 
be accepted without question. 

Oil-engines. — A volatile oil like naphtha or gasoline, 
which can be readily vaporized at ordinary temperatures, can 
be used in any gas-engine. It is sufficient to draw a part of 
the air-supply through some simple vaporizer and to use this 
mixture of air and vapor in place of gas. But there is danger 
that such volatile oils may take fire accidentally or may leak 
out or escape from tanks and reservoirs and mingle with air 
in confined spaces and form an explosive mixture. Though 
engines of some size have been run with gasoline, the applica- 
tion of such volatile oils to explosive engines appears to be 
best adapted to motor-carriages or to small open boats where 
there is less danger from the formation of explosive mixtures 
with air in confined spaces. Insurance companies in America 
and the government in England interfere with the storing and 
use of gasoline and other volatile oils, and so have checked 
their use in gas-engines. 

Much attention has been given in England to the develop- 
ment of engines which can use refined burning oils like kero- 
sene. The difficulty in the use of such oils is that they cannot 
be entirely vaporized by the application of heat alone even 
though raised to 6oo° F. If heated under pressure they 
decompose, and yield, together with a light volatile product, 
another product which is heavier than the original oil. If 
they are brought to a very high temperature, by passing over 
a red-hot surface for example, they are likely to yield car- 
bonaceous or tarry matter which clogs the vaporizer. The 
only exception appears to be the oil which is distilled from 
Scotch shales; it can be entirely vaporized at about 570 F., 
which may be due to the fact that it is made by a destructive 
distillation. 

The usual way of treating kerosene and other safe oils 



HOT- AIR ENGINES AND GAS-ENGINES. 2\J 

when they are used in engines is to spray or otherwise mingle 
them with hot air. Three ways have been devised for doing 
this. The first way was to spray the oil by an atomizer into 
an iron cylinder, which was heated by the exhaust gases while 
the engine was running. The air for the engine was drawn 
through this cylinder, and thus charged with oil in the form of 
vapor or perhaps partly in a spray or mist. In order to keep 
the vaporizing cylinder hot it was necessary to have an ex- 
plosion for every complete cycle, as the omission of an explosion 
would cool the vaporizer too much. The engine was governed 
by reducing simultaneously the oil and air-supply, which was 
found to be a very wasteful way. 

Another way of accomplishing the same thing is to have 
an extension to the cylinder which is not cooled by a water- 
jacket and which consequently becomes strongly heated. This 
extension, which has a constricted communication with the 
rest of the cylinder, is filled with hot spent gases after an 
explosion, into which the oil is sprayed and by which it is 
vaporized. The next charge of air is compressed into the 
extension to the cylinder, mingles with the hot gases in it, 
and an explosive mixture is formed which is ignited by the 
heat of the walls or the heat of compression when the proper 
proportions are attained. A variation of this device has the 
entire cylinder cooled with a water-jacket, but a closed recep- 
tacle smaller than the cylinder and open only at the forward 
end is attached to the cylinder-head. This receptacle is of 
course heated by the explosions, and can vaporize the oil 
which is injected into the hot gases remaining in it after the 
exhaust. 

The third and perhaps the best way is to strongly heat 
a part of the air and draw it through oil in a hot vaporizer. 
This method brings the engine into much the condition of an 
engine using gasoline, and the usual way of governing by 
omitting explosions can be used. It is true that the preced- 
ing method may be governed by omitting explosions, but 
there is danger that the hot chamber may be too much cooled 



2l8 THERMODYNAMICS OF THE STEAM-ENGINE. 

if several explosions are omitted, and that the engine will 
stop. 

All methods involving the use of hot air or hot gas for 
vaporizing the oil have the defect that the charge is at a high 
temperature before it is compressed. The compression must 
not be carried too far for fear of a premature explosion, and 
again, the rise of temperature is less than for a cool charge. 
Consequently the efficiency of the engine is less than that of 
a gas-engine. 

The ideal way of treating safe oils for use in an explosive 
engine appears to be to decompose them in such a way as to 
get volatile products only. In refining petroleum and making 
kerosene it is customary to treat some of the heavier products 
by distillation and redistillation under pressure, and thus to 
gain a larger proportion of illuminating-oil. If a similar 
method could be used with kerosene, and thus a volatile 
product like gasoline could be made continuously as demanded 
by the engine, the main difficulty with the use of safe oils in 
explosive engines would be removed. 

Ignition. — After the mixture of air and oil is compressed 
in the cylinder of an explosion-engine it must be ignited. 
There are four ways of accomplishing this: (i) by an electric 
spark, (2) by the flame method, (3) by a heated tube, and 
(4) by compression in a hot chamber. 

The electric method requires a galvanic battery, an in- 
duction-coil, and a circuit-breaker — apparatus requiring some 
care and intelligence. It appears to be the favorite method 
in America, but is not liked in England. Older forms of this 
device had simply two points at the proper distance across 
which a spark leaped when the circuit was closed or broken by 
a circuit-breaker outside of the cylinder. The later form has 
the circuit-breaker inside the cylinder consisting of two points 
of platinum, one of which is pressed against or wiped over the 
other at the proper time by a gear driven by the engine. A 
spark is formed both when the circuit is closed and when it is 
broken ; the charge is usually fired by the spark which is 



HOT-AIR ENGINES AND GAS-ENGINES. 2IO, 

formed when the circuit is opened as it is the more intense. 
This apparatus is made in a substantial way and does not 
appear to fail or give trouble in any way. 

The flame ignition consists in conveying an ignited mass 
of gas and air from an igniting flame outside the cylinder of 
the engine to the explosive mixture inside. The space or 
cavity in which the burning gas is conveyed is usually in a 
slide which may or may not serve as the valve for charging 
and exhausting the cylinder. To convey the burning gas to 
the charge the cavity must be partially filled with gas; the 
gas is ignited at a small jet burning steadily near the cylinder; 
the cavity is then closed and a small supply of explosive gas 
from the cylinder is let in to raise the pressure in the cavity 
lest the sudden opening of the cavity to the cylinder should 
extinguish the flame; the cavity, where the pressure is still 
less than in the cylinder, is brought to the explosion-port and 
the charge is fired ; the cavity must finally be ventilated to be 
ready for the next supply of gas. This method, using 
illuminating gas, works well up to 80 ignitions a minute. 
Clerk has a flame ignition in which the flame cavity is fed only 
from the compressed explosive gas in the cylinder. The 
cavity consequently does not require ventilation and is more 
quickly filled with gas. He has used it to make 300 ignitions 
a minute. 

The hot tube requires only a small iron tube, which is kept 
red-hot by a Bunsen burner or other heating flame. The 
tube comes out horizontally from the cylinder and sometimes 
is turned upward for convenience in heating. At the proper 
time the explosive mixture in the cylinder is admitted to the 
tube by a valve which is worked by the engine. Sometimes 
the tube has an inlet-valve at the outer end to ventilate the 
tube with air drawn in during the filling stroke. 

In the discussion of oil-engines reference has already been 
made to the fact that the charge may be exploded sponta- 
neously by compressing it into a hot receptacle. In some 
engines this occurs when the hot gas with its charge of oil in 



220 THERMODYNAMICS OF THE STEAM-ENGINE. 

the receptacle is mingled with a proper amount of air to form 
an explosive mixture. In other engines the explosion takes 
place when the mixture of air and oil is compressed to a 
pressure which will cause an explosion in contact with the hot 
sides of the receptacle or with some other hot metallic surface. 
Considerable ingenuity has been shown in properly propor- 
tioning oil-engines so that the explosion shall take place 
spontaneously at the proper time. This method is better 
adapted to oil-engines than to gas-engines, as a mixture of oil 
and air is more readily exploded and at a lower temperature 
than a mixture of gas and air. 

Gas-engine Governors. — The only effective way of gov- 
erning a gas-engine is to omit an explosion from time to time, 
so that the engine shall run at less than full power. When 
the engine is running light two or more explosions in succes- 
sion may be omitted. When the engine attains too high a 
speed the governor draws back the cam which opens the gas- 
valve and no gas is admitted; the engine consequently takes 
in air only, compresses and expands and then exhausts it, but 
no work is developed by the engine. The advantage of this 
method is that the engine is as efficient at small as at large 
loads; owing to the clearing of the cylinder of spent gases 
during the cycle when no explosion takes place the succeed- 
ing working cycle is more efficient. The disadvantage is that 
the engine runs irregularly, as the only source of energy when 
the engine fails to explode is the fly-wheel. This, and the 
fact that only one stroke in four is a working stroke, explains 
the use of heavy fly-wheels for explosive engines. 

The reason why the engine must be governed by omitting 
an explosion is that the proportion of the mixture of gas and 
air can be varied only within narrow limits if it is to explode 
when ignited. Some engines for electric lighting have been 
made to diminish the gas-supply when the load is smaller than 
full power, but the variation of power by this means is very 
small and must be supplemented by omitting explosions if the 
load is much diminished. Large gas-engines with two cylinders 



HOT-AIR ENGINES AND GAS-ENGINES. 



221 



are of course more easily controlled than single-cylinder 
engines, since they have an explosion for every revolution. 

The Diesel Motor. — A new form of internal-combustion 
engine was described by Rudolf Diesel in 1893, which does 
away with many of the difficulties of gas- and oil-engines and 
which at the same time gives a much higher efficiency. The 
essential feature of his engine consists in the adiabatic com- 
pression of atmospheric air to a sufficient temperature to 
ignite the fuel which is injected at a determined rate during 
part of the expansion or working stroke. The first experi- 
mental engines were built at Augsburg in 1894 and 1896, one 
of 12 and another of 20 horse-power; now the construction 
of these engines is undertaken at several places in Europe and 
in America. 

The cycle of all of the engines thus far built under 
Diesel's patents is represented by Fig. 50, which represents 
four strokes of a single-acting piston 
or plunger. Atmospheric air is 
drawn in from a to b and is com- 
pressed from b to c to a pressure of 
500 pounds to the square inch and a 
temperature of 1000 F. From c to 
d fuel is injected in a finely divided 
form, and as there is air in excess 
it burns completely at a rate that 
can be controlled by the injection 
mechanism. Thus far the only fuel 
used is petroleum or some other oil. 
At d the supply of fuel is interrupted Fig 50. 

and the remainder of the working stroke, de, is an adiabatic 
expansion. The cycle is completed by a release at e and a 
rejection of the products of combustion from b to a. 

The cycle has a resemblance to that of the Otto engine, 
but differs in that the air only is compressed in the cylinder 
and the combustion is accompanied by an expansion. Diesel, 
in his theoretic discussion of his engine, stipulates that the 




222 THERMODYNAMICS OF THE STEAM-ENGINE. 

rate of combustion shall be so regulated that the temperature 
shall not rise during the injection of fuel, and that the line cd 
shall therefore be very nearly an isothermal for a perfect gas. 
Since the fuel is added during the operation represented by 
the line cd, the weight of the material in the cylinder increases 
and its physical properties change, so that the line will not 
be a true isothermal. The fact that there is air in excess 
makes it probable that these changes of weight and properties 
will be insignificant. On the other hand, it is not probable 
that in practice the rate of injection of fuel will be regulated 
so as to give no rise of temperature, or that there is any great 
advantage in such a regulation if the temperature is not 
allowed to rise too high. 




Fig. 51. 

The diagram from an engine of this type is shown by 
Fig. 51, which appears to show an introduction of fuel for 
one-eighth or one-seventh of the working stroke. It is prob- 
able that the compression and the expansion after the cessa- 
tion of the fuel supply are not really adiabatic, though as 
there is nothing but dry gas in the cylinder during those 
operations the deviation may not be large. The sides and 
heads of the cylinders of all the engines thus far constructed 
are water-jacketed, though the use of such a water-jacket and 
the consequent waste of heat was one of the difficulties in the 
use of internal-combustion engines that Diesel sought to 
avoid by controlling the rate of combustion. 



HOT-AIR ENGINES AND GAS-ENGINES. 22$ 

The oil used as fuel is injected in form of a spray by air 
that is compressed separately in a small pump to 30 or 40 
pounds pressure above that in the main cylinder; of course it 
is necessary to cool this portion of the air after compression 
to avoid premature ignition. The engines that have been 
used are described as giving a clear and nearly dry exhaust. 
In damp weather the exhaust shows a little moisture, prob- 
ably from the combustion of hydrogen in the oil. The 
cylinder when opened shows a slight deposit of soot on the 
head. It appears therefore that Diesel has succeeded in 
constructing an engine for burning heavy oils with good 
economy and without the annoyances of an igniting device. 
The engines have the further advantage in that the work can 
be regulated by the amount of fuel supplied, which amount is 
not controlled, as in explosive engines, by the necessity to 
form an explosive mixture. The discussion of the theoretical 
efficiency of the cycle shows that the efficiency increases as 
the time of injection of fuel is shortened. In practice the 
engine shows a slight decrease in economy for light loads, due 
probably to the losses by radiation and to the water-jacket, 
which are nearly constant for all loads. 

In the exposition of the theory of his motor, Diesel * 
claims that all kinds of fuel, solid, liquid, and gaseous, can be 
burned in his motor. As yet oil only has been used; the 
choice of petroleum or other heavy oil has probably been due 
to the low cost of such oils. It is evident that gas may be 
used in this type of engine ; the gas can be compressed 
separately to a pressure somewhat higher than that in the 
main cylinder, much as the air is which is used for injecting 
oil. It does not appear necessary to cool the gas after com- 
pression, as it will burn only when supplied with air. 

There appears to be no insurmountable difficulty in sup- 
plying powdered solid fuel to this engine. The presence of 
the ash from such fuel in the cylinder may, however, be 
expected to give trouble. Diesel claims that with a large 

* Rational Heat Motor; Rudolf Diesel, trans. Bryan Donkin. 



224 THERMODYNAMICS OF THE STEAM-ENGINE, 

excess of air (for example, a hundred pounds of air for one 
pound of coal) the ash will be swept out of the cylinder with 
the spent gases and will not give trouble; but that claim is 
accompanied by an assumption that air may be compressed 
isothermally if water is injected into the cylinder of the com- 
pressor, and one idea appears to be as improbable as the 
other. 

All the engines thus far built have single-acting pistons or 
plungers which perform all four operations for a cycle in one 
cylinder. Some of the engines have three cylinders with 
their plungers connected to a three-throw crank-shaft; but in 
that case each plunger has its own complete cycle performed 
in four strokes, just as for the single-cylinder engine. Diesel 
has, however, described a compound engine which has both 
compression and expansion divided into two stages. Such 
an engine may have a compression-pump, a combustion- 
cylinder, and an expansion-cylinder. The pump takes air 
from the atmosphere and compresses it to a moderate pressure 
and delivers it to a receiver. The combustion-cylinder has a 
plunger like that of the simple engine which performs a cycle 
in four strokes. It takes air at nearly constant pressure from 
the receiver on the first down stroke and compresses it to the 
proper pressure and temperature to ignite the fuel on the fol- 
lowing up stroke; during a part of the second down stroke 
fuel is injected at such a rate as to produce isothermal expan- 
sion, and the remainder of that stroke gives adiabatic ex- 
pansion with a reduction of temperature; the products of 
combustion cooled to a considerable extent by the partial 
adiabatic expansion in the combustion-cylinder are transferred 
to the expansion-cylinder during the fourth stroke of the 
plunger, which completes the cycle of the combustion-cylinder. 
The proportions of the three cylinders should be such that 
the fourth stroke of the plunger just mentioned should have 
for its terminal pressure the pressure in the receiver. The 
expansion-cylinder completes the expansion of the products 
of combustion and then discharges them into the atmosphere; 



HOT-AIR ENGINES AND GAS-ENGINES, 22$ 

by its use the final volume of those gases when release occurs 
may be greater than the volume of the air drawn in by the 
compression-pump, and if desired, the expansion may be car- 
ried down to the pressure of the atmosphere. By assuming 
an isothermal compression in the compression-pump, Diesel 
works out a theoretic cycle which approaches very closely in 
appearance and efficiency to Carnot's cycle for a perfect gas; 
it appears that his attention was first directed to the motor he 
has invented by an attempt to describe an engine which could 
w r ork on Carnot's cycle. But he assumes that an isothermal 
compression can be obtained by simply spraying water into 
the pump-cylinder during compression, which is contrary to 
experience in the use of air-compressors. Even if such an 
isothermal compression were practicable, it is doubtful whether 
it would be desirable ; as it would require a much higher 
pressure to produce the proper ignition temperature in the 
combustion-cylinder than is now employed in single-cylinder 
engines; but the high pressure required to give ignition is one 
of the principal sources of difficulty in the single engine. 

Since both compression-pump and expansion-cylinder, 
even with single-acting pistons, will have one working stroke 
for each revolution, while the combustion-cylinder can give a 
working stroke for two revolutions, it is clear that the com- 
pound engine must have at least two combustion-cylinders 
with their plungers working together, but having their opera- 
tions half a cycle apart. The expansion-piston should have 
its crank opposite those of the combustion-plungers. The 
compression-pump may have its crank set at any convenient 
angle, since it delivers air to a reservoir. 

A theoretical discussion of the efficiency of the cycle for 
the simple engine as represented by Fig. 50 may be obtained 
by considering that heat is added at constant temperature 
from e to d and that heat is rejected at constant volume from 
e to by bearing in mind that be and dc represent adiabatic 
changes. 

From equation (75), page 65, the expression for the heat 



226 THERMODYNAMICS OF THE STEAM-ENGINE. 

supplied from c to d is, for one pound of working substance, 

(2, = A Pc v c log,-* = ART C log, -*. 
The heat rejected at constant volume is 

& = c v (T e - T b ) = c -i(T e - T b ). 
Since the expansion de is adiabatic, 

t. = ri^f " ' = tI 

\vj v 

but since the compression fc is also adiabatic, 



and consequently 

r.= ri5)- I (5)- , = r i (5p .. 

for v e = b b . Replacing T e by its value in the expression for 
Q %i we have 

K ( W 

Finally, the efficiency appears to be 

a- a , ^ "' 

7; — — =1 



Inspection of the equation shows that the efficiency may 
be increased by raising the temperature T c or by reducing the 
temperature T b . The latter is practically the temperature of 
the atmosphere, but T c may be made any desired temperature 
by reducing the clearance of the cylinder and thus raising the 
pressure at the end of compression. Again, the efficiency 



HOT-AIR ENGINES AND GAS-ENGINES. 227 

may be increased by reducing the time during which fuel is 
injected, that is, by reducing the ratio v d : v cJ as may be 
proved by a series of calculations with different values for that 
ratio. This is a very important conclusion, as it shows that 
the engine will have in practice little if any falling off in 
efficiency at reduced loads. 

It is reported that a clearance of something less than 7 per 
cent is associated with a compression to 500 pounds and a 
temperature of 1000 F., or more. Taking the pressure of 
the atmosphere at 14.7 pounds per square inch, adiabatic 
compression to 500 pounds above the atmosphere or to 514.7 
pounds absolute requires a clearance of 

1 1 

so that the clearance is 

0.0796 -v- { 1 — 0.0796 } = 0.0865 

of the piston displacement. 

If the temperature of the atmosphere be taken at 70 F. 
or 530.7 absolute, the temperature after adiabatic compres^ 
sion becomes 

^=4 : )- = 53 o.;(i^)^= I4 So 

absolute, or 1020 F. 

If it be further assumed that fuel is supplied for one-tenth 
of the working stroke, then 

v d = 0.1(^-^)1^ = 0.1(1 — 0.0796) -f- 0.0796^ 

= 0.1716^. 

The equation for efficiency gives in this case 



228 THERMO D YNAMICS OF THE STEAM-ENGINE. 

778 X 0.2375 X 530.7 M-g) - 1 

*=!-■ O.I7I6 =a58 ' 

1.405 X 53-22 X 1480 log, — i_ 

0.0796 

An engine giving 26.6 indicated horse-power (cheval-a- 
vapeur) and exerting 19.2 horse-power at the brake is reported 
to consume 223 grams of petroleum per hour and to give a 
thermal efficiency of 28.6 per cent. Now if the heats of 
combustion of coal and petroleum are taken at 1400 B. T. u. 
and 20,000 B. T. U., respectively, one gram of petroleum is 
equivalent to -^ of a gram of coal, and the performance of 
the Diesel engine is equal to the consumption of 333 grams 
of coal per horse-power per hour, or to about f of a pound of 
coal per English horse-power per hour. To make the com- 
parison complete the relative costs of coal and petroleum 
should be considered, together with the probability that a 
large demand for petroleum would be liable to affect its price. 



CHAPTER XI. 



THE STEAM-ENGINE. 



THE steam-engine is, at the present time, the most im- 
portant heat-engine. When of large size and properly 
designed and managed it has at least as good economy as 
any other heat-engine, though it may be excelled in this 
regard by explosive gas-engines when they are fully devel- 
oped. It can be controlled, regulated, and reversed easily 
and positively — properties which are not possessed in like 
degree by other heat-engines. It is interesting to know that 
the theory of thermodynamics was developed mainly to 
account for the action and to provide methods of designing 
steam-engines; though neither object is entirely accom- 
plished, on account of the fact that the engine-cylinder must 
be made of some metal to be hard and strong enough to 
endure service, for all metals are good conductors of heat and 
seriously affect the action of a condensable fluid like steam. 

Carnot's Cycle for a steam-engine is represented by Fig. 
52, in which ab and cd are isothermal lines 
representing the application and rejection 
of heat at constant temperature and at 
constant pressure, be and da are adiabatic 
lines, representing change of temperature 
and pressure, without transmission of heat 
through the walls of the cylinder. The 
diagram representing Carnot's cycle has 
an external resemblance to the indicator- FlG - 52 - 

diagram from some actual engines,. but it differs in essential 

particulars. 

229 



a 




b 


1 


d 


— _ c 













230 THEE MOD YNAMICS OF THE STEAM-ENGINE. 

In the condition represented by the point a the cylinder 
contains a mixture of water and steam at the temperature t x 
and the pressure^. If connection is made with a source of 
heat at the temperature t x , and heat is added, some of the 
water will be vaporized and the volume will increase at con- 
stant pressure as represented by ab. If thermal communica- 
tion is now interrupted adiabatic expansion may take place as 
represented by be till the temperature is reduced to 2f 2 , the 
temperature of the refrigerator, with which thermal com- 
munication may now be established. If the piston is forced 
toward the closed end of the cylinder some of the steam in it 
will be condensed, and the volume will be reduced at constant 
pressure as represented by cd. The cycle is completed by an 
adiabatic compression represented by da. 

If the absolute temperature of the source of heat is T l9 
and if that of the refrigerator is 7" 2 , then the efficiency is 

T — T 

M i x a 

v= — r — i 

whatever may be the working fluid. 

For example, if the pressure of the steam during isother- 
mal expansion is ioo pounds above the atmosphere, and if the 
pressure during isothermal compression is equal to that of the 
atmosphere, then the temperatures of the source of heat and 
of the refrigerator are 337°. 6 F. and 2 12° F., or 798.4 and 
672.7 absolute, so that the efficiency is 

798.4 — 672.7 

— - — - = 0.157. 
798.4 D/ 

The following table gives the efficiencies worked out in a 
similar way, for various steam-pressures, — both for t^ equal to 
212 F., corresponding to atmospheric pressure, and for / 9 
equal to 116 F., corresponding to an absolute pressure of 
1.5 pounds to the square inch: 



THE STEAM-ENGINE. 23 1 

EFFICIENCY OF CARNOT'S CYCLE FOR STEAM-ENGINES. 



Initial Pressure 






by the Gauge, 


Atmospheric 


1.5 Pounds 


above the 


Pressure. 


Absolute. 


Atmosphere. 






15 


O.053 


O.189 


30 


O.084 


O.215 


60 


O.I24 


O.249 


IOO 


0.157 


O.278 


I50 


O.I86 


O.302 


200 


O.207 


O.320 


300 


O.238 


0-347 



The column for atmospheric pressure may be used as a 
standard of comparison for non-condensing engines, and the 
column for 1.5 pounds absolute may be used for condensing 
engines. 

It is interesting to consider the condition of the fluid in 
the cylinder at the different points of the diagram for Carnot's 
cycle. Thus if the fluid at the condition represented by b 
in Fig. 52 is made up of x b part steam and 1 — x b part water, 
then from equation (146) the condition at the point c is given 
by 

Xc^-^fext + Ot-e). . . . (244) 



In like manner the condition of the mixture at the point d is 



T ir 

x d = y [-f x * + i — ^ 



- • ( 2 45) 



It is interesting to note that if x b is larger than one-half, 
that is, if there is more steam than water in the cylinder at b, 
then the adiabatic expansion is accompanied by condensation. 
Again, if x a is less than one-half, then the adiabatic compres- 
sion is also accompanied by condensation. Very commonly 
it is assumed that x b is unity, so that there is dry saturated 
steam in the cylinder at b; and that x a is zero so that there 
is water only in the cylinder at a\ but there is no necessity 
for such assumptions, and they in no way affect the efficiency. 



232 THERMODYNAMICS OF THE STEAM-ENGINE. 

If the cylinder contains M pounds of steam and water, the 
heat absorbed by the working substance during isothermal 
expansion is 

Q x = Mr x (x b - x a ), (246) 

and the heat rejected during isothermal compression is 

<2 a = Mrlx c — x d ), (247) 

so that the heat changed into work during the cycle is 

2, - Q, = M\r x {x a - x b ) - rlx c - x d )}. . (248) 
But from equations (244) and (245) 

rj^Xc — x d ) = -^r,{x b — x a ), . . . (249) 

and the expression for the heat changed into work becomes 

G, - <2, = Mr x (x b - x a ) x ~ \ . . (250) 

This equation is deduced because it is convenient for making 
comparisons of various other volatile liquids and their vapors, 
with steam, for use in heat-engines. It is of course apparent 
that 

Q, - & T x -T % 
77 ~ Q x ~ T x ' 

from equations (250) and (246), a conclusion which is known 
independently, and indeed is necessary in the development 
of the theory of the adiabatic expansion of steam. 

In the discussion thus far it has been assumed that the 
working fluid is steam, or a mixture of steam and water. But 
a mixture of any volatile liquid and its vapor will give 
similar results, and the equations deduced can be applied 



THE STEAM-ENGINE. 233 

directly. The principal difference will be due to the 
properties of the vapor considered, especially its specific 
pressures and specific volumes for the temperatures of the 
source of heat and the refrigerator. 

For example, the efficiency of Carnot's cycle for a fluid 
working between the temperatures 160 C. and 40 C. is 

160 — 40 

-2 : — — = 0.277. 

l6o + 273.7 

The properties of steam and of chloroform at these tem- 
peratures are 

Steam. Chloroform. 

40 C 160 C. 40 C 160° C 

Pressure, mm. mercury 54-91 4651.4 369.26 8734.2 

Volume, cubic metres 19-74 0.3035 0.4449 0.0243 

Heat of vaporization, r 578-7 494-2 °3-i3 50-53 

Entropy of liquid, Q 0.1364 0.4633 0.03196 o. 11041 

For simplicity, we may assume that one kilogram of the 
fluid is used in the cylinder for Carnot's cycle, and that x b is 
unity while x a is zero, so that from equation (250) 

T x — T u 
■*■ 1 

and for steam 

Q x — Q* — 494-2 X 0.277 = 137 calories, 
while for chloroform 

Q x — Q, = 50-53 X 0.277 = H calories. 

After adiabatic expansion the qualities of the fluid will be, 
from equation (244), for steam 

40 4- 273.7/ 494-2 , ~ \ 

x c = o xx - h °-4 6 33 — 0.1364 = 0.795, 

578.7 V160+ 273.7 / 



234 THERMODYNAMICS OF THE STEAM-ENGINE. 

and for chloroform 



40 + 273.7/ 50.53 



63 



^ , + o. 11041 — 0.03196] = 0.960. 

.13 \i6o + 273.7 n ^ / 



The specific volumes after adiabatic expansion are, conse- 
quently, for steam 

v c — x c u^ + g = 0.795(19.74 — 0.001) + 0.001 = 15.7, 

and for chloroform 

V c — X c U^-\- <T = 0.969(0.4449 — O.OOO655) -f- O.OOO655 = 043I. 

These values for v c just calculated are the volumes in the 
cylinder at the extreme displacement of the piston, on the 
assumption that one kilogram of the working fluid is vaporized 
during isothermal expansion. A better idea of the relative 
advantages of the two fluids will be obtained by finding the 
heat changed into work for each cubic metre of maximum 
piston-displacement, or for a cylinder having the volume of 
one cubic metre. This is obtained by dividing Q x — (2 2 > the 
heat changed into work for each kilogram by v c . For steam 
the result is 

(2, - Q.) -*- *c = 137 -*- 157 = 8.73, 
and for chloroform it is 

(G, - (2.) -*- v c = 14 -*- 0.413 = 34 ; 

from which it appears that for the same volume chloroform 
can produce more than three and a half times as much power. 
Even if we consider that the difference of pressure for chloro- 
form, 

8734.2 — 369.3 = 8364.9 mm., 
is nearly twice that for steam, which has only 
4651.4 - 54.9 = 459 6 -5 mm- 



THE STEAM-ENGINE. 235 

difference of pressure, the advantage appears to be in favor 
of chloroform. If, however, the difference of pressures given 
for chloroform is allowable also for steam, giving 

8364.9 -\- 54.9 = 8419.8 mm. 

for the superior pressure, then the initial temperature for 
steam becomes i84°.9 C, and the efficiency becomes 

i84°.9 — 40 o 

= 0.318, 



184.9 + 273.7 



instead of 0.277. On the whole, steam is the more desirable 
fluid, even if we do not consider the inflammable and poison- 
ous nature of chloroform. Similar calculations will show that 
on the whole steam is at least as well adapted for use in heat- 
engines as any other saturated fluid; in practice, the cheap- 
ness and incombustibility of steam indicate that it is the 
preferable fluid for such uses. 

Non-conducting Engine. — The conditions required for 
alternate isothermal expansion and adiabatic expansion cannot 
be fulfilled for Carnot's cycle with steam any more than they 
could be for air. The diagram for the cycle with steam, 
however, is well adapted to production of power; the con- 
trary is the case with air, which gives a much attenuated 
diagram. 

In practice steam from a boiler is admitted to the cylinder 
of the steam-engine during that part of the cycle which 
corresponds to the isothermal expansion of Carnot's cycle, 
thus transferring the isothermal expansion to the boiler, where 
steam is formed under constant pressure. In like manner 
the isothermal compression is replaced by an exhaust at con- 
stant pressure, during which steam may be condensed in a 
separate condenser, cooled by cold water. 

By proper valve-gear the expansion and compression of 
Carnot's cycle may be simulated, thus giving a diagram hav- 
ing an external resemblance to Carnot's cycle. The cylinder 




236 THERMODYNAMICS OF THE STEAM-ENGINE. 

is commonly made of cast iron, and is always some kind of 
metal; there is consequently considerable interference due to 
the conductivity of the walls of the cylinder, and the expan- 
sion and compression are never adiabatic. There is an advan- 
tage, however, in discussing first an engine with a cylinder 
made of some non-conducting material, although no such 
material proper for making cylinders is now known. 

The diagram representing the operations in a non-con- 
ducting cylinder for a steam-engine can be represented by 
Fig* 53- a b represents the admission of 
dry saturated steam from the boiler; be is 
an adiabatic expansion to the exhaust 
pressure ; cd represents the exhaust ; and 
da is an adiabatic compression to the initial 
Fig. 53. pressure. It is assumed that the small 

volume, represented by a, between the piston and the head 
of the cylinder is filled with dry steam, and that the steam 
remains homogeneous during exhaust so that the quality is 
the same at d as at c. These conditions are consistent and 
necessary, since the change of condition due to adiabatic 
expansion (or compression) depends only on the initial condi- 
tion and the initial and final pressures; so that an adiabatic 
expansion from a to d would give the same quality at d as 
that found at c after adiabatic expansion from b, and con- 
versely adiabatic compression from d to a gives dry steam at 
a as required. 

The cycle represented by Fig. 53 differs most notably 
from Carnot's cycle (Fig. 52) in that ab represents admission 
of steam and cd represents exhaust of steam, as has already 
been pointed out. It also differs in that the compression da 
gives dry steam instead of wet steam. The compression line 
da is therefore steeper than for Carnot's cycle, and the area of 
the figure is slightly larger on this account. This curious fact 
does not indicate that the cycle has a higher efficiency; on 
the contrary, the efficiency 7 is less, and the cycle is incom- 
plete. To complete the cycle for a non-conducting cylinder 



THE STEAM-ENGINE. 2^,7 

the exhausted steam must be condensed, pumped back into 
the boiler, and reevaporated. 

If the pressure during admission (equal to the pressure in 
the boiler) is/,, and if the pressure during exhaust is/ a , then 
the heat required to raise the water resulting from the con- 
densation of the exhaust-steam is 

?i — ?*> 

where q x is the heat of the liquid at the pressure p l and q 2 is 
the heat of the liquid at the pressure / 2 . The heat of vapori- 
zation at the pressure/, is r lt so that the heat required to raise 
the feed-water from the temperature of the exhaust to the 
temperature in the boiler and evaporate it into dry steam is 

& = ' , i + & — ?■; ( 2 5 

and this is the quantity of heat supplied to the cylinder per 
pound of steam. 

The steam exhausted from the cylinder has the quality x 2 
calculated by aid of the equation 

*. = -*(£ - 

r, 



and the heat that must be withdrawn when it is condensed is 

0t = *v.; (252) 

this is the heat rejected from the engine. The heat changed 
into work per pound of steam is 

Qx — Q* = r > + Qx - ?. — *W • • • ( 2 53) 

But part of the work is used in pumping the condensed 
steam or feed-water into the boiler from the pressure /, to 
the pressure/,. The heat equivalent of this work is 

^,-AK (254) 

where cr is the specific volume of water. This heat should 
be subtracted from the heat changed into work per pound of 



238 THERMODYNAMICS OF THE STEAM-ENGINE. 

steam to find the available energy developed by the engine. 
But the heat represented by the expression (254) is small 
compared with the heat changed into work as represented by 
equation (253), and may be neglected. 
The efficiency of the cycle is 

V = ^^=I-—7^ • • • • (255) 

If values are assigned to/ t and/, and the proper numeri- 
cal calculations are made, it will appear that the efficiency for 
a non-conducting engine is always less than the efficiency for 
Carnot's cycle between the corresponding temperatures. 

It should be remarked that the efficiency is not affected 
by the clearance or space between the piston and the head of 
the cylinder and the space in the steam-passages of the 
cylinder, provided that the clearance is filled with dry sat- 
urated steam as indicated in Fig. 52. This is evident from 
the fact that no term representing the clearance, or volume 
at a, Fig. 52, appears in equation (254). Or, again, we may 
consider that the steam in the cylinder at the beginning of 
the stroke, occupying the volume represented by a, expands 
during the adiabatic expansion and is compressed again dur- 
ing compression, so that one operation is equivalent to and 
counterbalances the other, and so does not affect the efficiency 
of the cycle. 

Incomplete Cycle. — The cycle for a non-conducting 
engine may be incomplete because the expansion is not 

carried far enough to reduce the pres- 
sure to that of the back-pressure line, 
as is shown in Fig. 54. Such an in- 
complete cycle has less efficiency than 
a complete cycle, but in practice the 
advantage of using a smaller cylinder 
Fig. 54- anc [ Q f reducing friction is sufficient 

compensation for the small loss of efficiency due to a moderate 
drop at the end of the stroke, as shown in Fig. 54. 




THE STEAM-ENGINE. 239 

The discussion of the incomplete cycle is simplified by- 
assuming that there is no clearance and no compression as is 
indicated by Fig. 54. It will be shown later that the 
efficiency will be the same with a clearance, provided the 
compression is complete. 

The most ready way of finding the efficiency for this cycle 
is to determine the work of the cycle. Thus the work dur- 
ing admission is 

A(«i + °")» (25Q 

where u x is the increase of volume due to vaporization of a 
pound of steam and <r is the specific volume of water. The 
work during expansion is 



E b - E c = -(p x + q x — x c p c - q c ), . . . (257) 



where q x and p x are the heat of the liquid and the heat-equiv- 
alent of the external work during vaporization at the pressure 
p x , while q c and p c are corresponding quantities for the 
pressure at c. x c is to be calculated by the equation 



*- = T& + > - 4 






The work done by the piston on the steam during ex- 
haust is 

p£x c u c + a). 

The total work of the cycle is obtained by adding the 
work during admission and expansion and subtracting the 
work during exhaust, giving 

i(p x + Ap x u x - x c p c - Ap,x c u c + q x - q c ) + (p x - p,)cr. (258) 



240 THERMODYNAMICS OF THE STEAM-ENGINE. 

The last term is small, and may be neglected. Adding and 
subtracting Ap c x c u c and multiplying by A, we get for the 
heat-equivalent of the work of the cycle 

Qx — C 9 = r x - x c r c -\-A(p c — p,)u c x c + ft — ft> • (259) 

which is equal to the difference between the heat supplied 
and the heat rejected as indicated. The heat supplied is 

(2, = r , + ft - ft, 

as was deduced for the complete cycle; the cost of making 
the steam remains the same, whether or not it is used effi- 
ciently. Finally, the efficiency of the cycle is 

„ _. 61 — Q% __ ^1 + ft - x c r e — q c + A{p c — p,)x c u c § 
0i r x + ft - q, 

. n—i- XcTc + ft - ft ~ ^( A - AM , 26q v 

^ + ft - ft 

If/, is made equal to/ a in the preceding equation, it will 
be reduced to the same form as equation (254), because the 
cycle in such case becomes complete. 

Steam-consumption of Non-conducting Engine. — A 
horse-power is 33000 foot-pounds per minute or 60 X 33000 
foot-pounds per hour. But the heat changed into work per 
pound of steam by a non-conducting engine with complete 
expansion is, by equation (253), 

^ + ft — ft*/",. 
so that the steam required per horse-power per hour is 

60 X 33000 
778(r 1 + ft - ft - xj$ 



THE STEAM-ENGINE. 24 1 

Similarly, the steam per horse-power per hour for an engine 
with incomplete expansion, by aid of expression (258), is 

60 X 33000 
77%{Pi + A Mi — x cf>c — Apjcji e + q x — g e ) 9 

The value of x^ or x c is to be calculated by the general equa- 
tion 

--Xt + "■->)■ 

The denominator in either of the above expressions for the 
steam per horse-power per hour is of course the work done 
per pound of steam and the parenthesis without the mechan- 
ical equivalent 778 is equal to Q x — Q 2 . If then we multiply 
and divide by 

Gi = *\+£, — ft. 

that is, by the heat brought from the boiler by one pound of 
steam, we shall have in either case 

60 x 33000 x 61 _ 60 x 33000 , 6 v 

778(0, - GOG, ~ 778^+ ?, - ?,)' ' ' { } 
where 

„_ g, -a 

is the efficiency for the cycle. 

Actual Steam-engine. — The indicator-diagram from an 
actual steam-engine differs from the cycle for a non-conduct- 
ing engine in two ways: there are losses of pressure between 
the boiler and the cylinder and between the cylinder and the 
condenser, due to the resistance to the flow of steam through 
pipes, valves, and passages; and there is considerable inter- 
ference of the metal of the cylinder with the action of the 
steam in the cylinder. The losses of pressure may be mini- 



242 THERMODYNAMICS OF THE STEAM-ENGINE. 

mized for a slow-moving engine by making the valves and 
passages direct and large. The interference of the walls of 
the cylinder cannot be prevented, but may be ameliorated by 
using superheated steam or by steam-jacketing. 

When steam enters the cylinder of an engine, some of it 
is condensed on the walls which were cooled by contact with 
exhaust-steam, thereby heating them up nearly to the tempera- 
ture of the steam. After cut-off the pressure of the steam is 
reduced by expansion and some of the water on the walls of 
the cylinder vaporizes. At release the pressure falls rapidly 
to the back-pressure, and the water remaining on the walls is 
nearly if not all vaporized. It is at once evident that so much 
of the heat as remains in the walls until release and is thrown 
out during exhaust is a direct loss; and again, the heat which 
is restored during expansion does work with less efficiency, 
because it is reevaporated at less than the temperature in the 
boiler or in the cylinder during admission, A complete state- 
ment of the action of the walls of the cylinder of an engine, 
with quantitative results from tests on engines, was first given 
by Hirn. His analysis of engine tests, showing the inter- 
changes of heat between the walls of the cylinder and the 
steam, will be given later. It is sufficient to know now that 
a failure to consider the action of the walls of the cylinder 
leads to gross errors, and that an attempt to base the design 
of an engine on the theory of a steam-engine with a non- 
conducting cylinder can lead only to confusion and disappoint- 
ment. 

The most apparent effect of the influence of the walls of 
the cylinder on the indicator-diagram is to change the expan- 
sion and the compression lines; the former exhibits this 
change most clearly. In the first place the fluid in the 
cylinder at cut-off consists of from twenty to fifty per cent 
hot water, which is found mainly adhering to the walls of the 
cylinder. Even if there were no action of the walls during 
expansion the curve would be much less steep than the adia- 
batic line for dry saturated steam. But the reevaporation 



THE STEAM-ENGINE. 243 

during expansion still further changes the curve, so that it is 
usually less steep than the rectangular hyperbola. 

It may be mentioned that the fluctuations of temperature 
in the walls of a steam-engine cylinder caused by the conden- 
sation and reevaporation of water do not extend far from the 
surface, but that at a very moderate depth the temperature 
remains constant so long as the engine runs under constant 
conditions. 

The performance of a steam-engine is commonly stated in 
pounds of steam per horse-power per hour. For example, a 
small Corliss engine, developing 16.35 horse-power when 
running at 61.5 revolutions per minute under 77.4 pounds 
"boiler-pressure, used 548 pounds of steam in an hour. The 
steam consumption was 

548+- 16.35 = 33-5 

pounds per horse-power per hour. 

This method was considered sufficient in the earlier his- 
tory of the steam-engine, and may now be used for comparing 
simple condensing or non-condensing engines which use sat- 
urated steam and do not have a steam-jacket, for the total 
heat of steam, and consequently the cost of making steam 
from water at a given temperature increases but slowly with 
the pressure. 

The performance of steam-engines may be more exactly 
stated in British thermal units per horse-power per minute. 
This method, or some method equivalent to it, is essential in 
making comparisons to discover the advantages of superheat- 
ing, steam-jacketing, and compounding. For example, the 
engine just referred to used steam containing two per cent of 
moisture, so that x y at the steam-pressure of 77.4 pounds was 
0.98. The barometer showed the pressure of the atmosphere 
to be 14.7 pounds, and this was also the back-pressure during 
exhaust. If it be assumed that the feed-water was or could 
be heated to the corresponding temperature of 212 F., the 



244 THERMODYNAMICS OF THE STEAM-ENGINE. 

heat required to evaporate it against 77.4 pounds above the 
atmosphere or 92.1 pounds absolute was 

Xf x -\-q^ — q^ = 0.98X888.4+ 291.7— 180.8 = 981.5 B. T. U* 

The thermal units per horse-power per minute were 

981.5 x 33.5 



60 



= 548. 



Efficiency of the Actual Engine. — When the thermal 
units per horse-power per minute are known or can be readily 
calculated, the efficiency of the actual steam-engine may be 
found by the following method: A horse-power corresponds 
to the development of 33000 foot-pounds per minute, which 
are equivalent to 

33000 -T- 778 = 42.42 

thermal units. This quantity is proportional to Q^ — Q„ and 
the thermal units consumed per horse-power per minute are 
proportional to Q lt so that the efficiency is 

= 61- ft 42.42 

^ ~ <2i ~ B - T - U. per H.P. per min.' 

For example, the Corliss engine mentioned above had an 
efficiency of 

42.42 -7- 548 = 0.077. 

This same method may evidently be applied to any heat- 
engine for which the consumption in thermal units per horse- 
power per hour can be applied. 

From the tests reported on page 327 it appears that the 
engine in the laboratory of the Massachusetts Institute of 
Technology on one occasion used 13.73 pounds of steam per 
horse-power per hour, of which 10.86 pounds were supplied 
to the cylinders and 2.87 pounds were condensed in steam- 



THE STEAM-ENGINE. 245 

jackets on the cylinders. The steam in the supply-pipe had 
the pressure of 157.7 pounds absolute, and contained 1.2 per 
cent of moisture. The heat supplied to the cylinders per 
minute in the steam admitted was 

lO.S6(x 1 r 1 -f- q x — q t ) -i- 60 

= 10.86(0.988 X 858.3 -f- 334- 2 — 126.4) -r-6o= 191. 1 B. T. u. ; 

q^ being the heat of the liquid at the temperature of the back- 
pressure of 4.5 pounds absolute. 

The steam condensed in the steam-jackets was withdrawn 
at the temperature due to the pressure and could have been 
returned to the boiler at that temperature; consequently the 
heat required to vaporize it was r v and the heat furnished by 
the steam in the jackets was 

2.87 X 0.98 X 858.3 ~ 60 = 40.6 B. T. U. 

The heat consumed by the engine was 

191. 1 +40.6 = 231.7 B.T. u. 

per horse-power per minute, and the efficiency was 

tf = 42.42-^- 231.7 = 0.183. 

The efficiency of Carnot's cycle for the range of tempera- 
tures corresponding to 157.7 and 4. 5 pounds absolute, namely, 
822°.9 and 6i8°.4 absolute, is 

T x —T t 822.9-618.4 _ 

v = ^rr = 822.9 -°' 2 ^' 

The efficiency for a non-conducting engine with complete 
expansion, calculated by equation (255), is for this case 

„ ^r 2 r 2 O.8214 X IOO4.2 
rf' = 1 — — = 1 — — j- 2 . — O.227 ; 

*x + 2i ~ q* 858.3 + 334.2 - 126. 1 



246 THERMODYNAMICS OF THE STEAM-ENGINE. 

where x t is calculated by the equation 

618.4/858.3 , \ 

= — I- 2 — ^ +0.5184 - 0.2278 = 0.8214. 

1004.2^822.9 ' ' I ^ 

During the test in question the terminal pressure at the 
end of the expansion in the low-pressure cylinder was 6 
pounds absolute, which gives 



TJr 

x c = - 



i + I - e ) 



630.8/858.2 . \ 

= ~ — br h 0.5184 — 0.2480 = 0.8317, 

995.2V822.9 ' J * J ° n 

and the efficiency by equation (260) was 



>\ + & — ft 

_ r 0.8317X995-2- 138.6+126.1+141(6-4.5)0.8317+61.65 

858.3 + 334.2 — 126.1 

= 0.222. 

The real criterion of the perfection of the action of an 
engine is the ratio of its actual efficiency to that of a perfect 
engine. If for the perfect engine we choose Carnot's cycle 
the ratio is 

V 0.183 - 

-, = ;r- = 0.736. 

rf 0.2485 /D 

But if we take for our standard an engine with a cylinder of 
non-conducting material the ratio for complete expansion is 

V - l8 3 

+ = - = 0.807. 

rf' 0.227 



THE STEAM-ENGINE. 2Afi 

For incomplete expansion the ratio is 

V 0.183 



T}'" ' 0.222 



O.824. 



To complete the comparison it is interesting to calculate 
the steam-consumption for a non-conducting steam-engine by- 
equation (261), both for complete and for incomplete expan- 
sion. For complete expansion we have 



60 X 33000 

/0 : 2 — = 10.5 pounds, 



778 X 0.227(858.3 + 334.2 - • 126.1 
and for incomplete expansion 

60 x 33000 

778 X 0.222(858.3 + 334.2 — 126. 1) 



= 10.7 pounds 



per horse-power per hour. 

But if these steanvconsumptions are compared with the 
actual steam-consumption of 13.73 pounds per horse-power 
per hour, the ratios are 

10.5 -~ 13.73 = 0-766 and 10.7 4- 13./3 = 0.783, 

which are very different from the ratios of the efficiencies. 
The discrepancy is due to the fact that more than a fourth of 
the steam used by the actual engine is condensed in the 
jackets and returned at full steam temperature to the boiler, 
while the non-conducting engine has no jacket, but is assumed 
to use all the steam in the cylinder. 

From this discussion it appears that there is not a wide 
margin for improvement of a well-designed engine running 
under favorable conditions. Improved economy must be 
sought either by increasing the range of temperatures (raising 
the steam-pressure or improving the vacuum), or by choosing 
some other form of heat-motor, such as the gas-engine. 

Attention should be called to the fact that the real 



248 



THERMODYNAMICS OF THE STEAM-ENGINE. 



criterion of the economy of a heat-engine is the cost of pro- 
ducing power by that engine. The cost may be expressed 
in thermal units per horse-power per minute, in pounds of 
steam per horse-power per hour, in coal per horse-power per 
hour, or directly in money. The expression in thermal units 
is the most exact, and the best for comparing engines of the 
same class, such as steam-engines. If the same fuel can be 
used for different engines, such as steam- and gas-engines, 
then the cost in pounds of fuel per horse-power per hour may 
be most instructive. But in any case the money cost must 
be the final criterion. The reason why it is not more fre- 
quently stated in reports of tests is that it is in many cases 
somewhat difficult to determine, and because it is affected by 
market prices which are subject to change. 

At the present time a pressure as high as 150 pounds 
above the atmosphere is used where good economy is expected. 
It appears from the table on page 233, showing the efficiency 
of Carnot's cycle for various pressures, that the gain in 
economy by increasing steam-pressure above 150 pounds is 
slow. The same thing is shown even more clearly by the 
following table: 

EFFECT OF RAISING STEAM-PRESSURE. 



Boiler- 
pressure 
by Gauge. 


Efficiency, 
Carnot's Cycle. 


Non-conducting Engine. 


Probable Performance, Actual 
Engine. 


Efficiency. 


B. T.U. per 
H.P. per 
Minute. 


B.T.U. per 
H.P. per 
Minute. 


Steam 

per H.P. 

per Minute. 


I50 
200 
300 


O.302 
O.320 
0-347 


O. 272 
O.288 
O. 306 


156 
147 
135 


195 
184 
169 


11. 5 

10.5 

9.6 



In the calculations for this table the steam is supposed to 
be dry as it enters the cylinder of the engine and the back- 
pressure is supposed to be 1.5 pounds absolute, while the 
expansion for the non-conducting engine is assumed to be 
complete. The heat-consumption of the non-conducting 



THE STEAM-ENGINE. 249 

engine is obtained by dividing 42.42 by the efficiency; thus 
for 150 pounds 

42.42 -r- 0.272 = 156. 

The heat-consumption of the actual engine is assumed to 
be one-fourth greater than that of the non-conducting engine. 
The steam-consumption is calculated by the reversal of the 
method of calculating the thermal units per horse-power per 
minute from the steam per horse-power per hour, and for 
simplicity all of the steam is assumed to be supplied to the 
cylinder. But an engine which shall show such an economy 
for a given pressure as that set down in the table must be a 
triple or a quadruple engine and must be thoroughly steam- 
jacketed. The actual steam-consumption is certain to be a 
little larger than that given in the table, as steam condensed 
in a steam-jacket yields less heat than that passed through 
the cylinder. 

It is very doubtful if the gain in fluid efficiency due to 
increasing steam-pressure above 150 or 200 pounds is not 
offset by the greater friction and the difficulty of maintaining 
the engine. Higher pressures than 200 pounds are used only 
where great power must be developed with small weight and 
space, as in torpedo-boats. 

Condensers. — Two forms of condensers are used to con- 
dense the steam from a steam-engine, known as jet-condensers 
and surface-condensers. The former are commonly used for 
land engines; they consist of a receptacle having a volume 
equal to one-fourth or one-third of that of the cylinder or 
cylinders that exhaust into it, into which the steam passes 
from the exhaust-pipe and where it meets and is condensed 
by a spray of cold water. 

If it be assumed that the steam in the exhaust-pipe is dry 
and saturated and that it is condensed from the pressure p 
and cooled to the temperature t k , then the heat yielded per 
pound of steam is 

^ — 4k* 



250 THERMODYNAMICS OF THE STEAM-ENGINE. 

where A is the total heat of steam at the pressure / and q k is 
the heat of the liquid at the temperature t k . The heat 
acquired by each pound of condensing or injection water is 

qi —qk, 

where gi is the heat of the liquid at the temperature t t of the 
injection-water as it enters the condenser. Each pound of 
steam will require 

A — q k 



G = 



qk — qi 



pounds of injection-water. 

For example, steam at 4 pounds absolute has for the total 
heat 1 128.6. If the injection-water enters with a temperature 
of 6o° F. and leaves with a temperature of 120 F., then 
each pound of steam will require 

X — q k 1 128.6 — 88.1 

q h - q { = 88.1 - 28.12 = I7 ' 3 

pounds of injection-water. This calculation is used only to 
aid in determining the size of the pipes and passages leading 
water to and from the condenser, and the dimensions of the 
air-pump. Anything like refinement is useless and impossi- 
ble, as conditions are seldom well known and are liable to vary. 
From 20 to 30 times the weight of steam used by the engine 
is commonly taken for this purpose. 

The jet-condensers cannot be used at sea when the boiler- 
pressure exceeds 40 pounds by the gauge ; all modern 
steamers are consequently supplied with surface-condensers 
which consist of receptacles, which are commonly rectan- 
gular in shape, into which steam is exhausted, and where it 
is condensed on horizontal brass tubes through which cold 
sea-water is circulated. The condensed water is drained out 
through the air-pump and is returned to the boiler. Thus 
the feed-water is kept free from salt and other mineral matter 
that would be pumped into the boiler if a jet-condenser were 



THE STEAM-ENGINE. 25 I 

used, and if the feed-water were drawn from the mingled 
water and condensed steam from such a condenser. Much 
trouble is, however, experienced from oil used to lubricate 
the cylinder of the engine, as it is likely to be pumped into 
the boilers with the feed-water, even though attempts are 
made to strain or filter it from the water. 

The water withdrawn from a surface-condenser is likely to 
have a different temperature from the cooling water when it 
leaves the condenser. If its temperature is t lt then we have 
instead of equation (261) 

G = ^^ (262) 

9k — <li 

for the cooling water per pound of steam. The difference is 
really immaterial, as it makes little difference in the actual 
value of G for any case. 

Cooling Surface. — Experiments on the quantity of cool- 
ing surface required by a surface-condenser are few and unsat- 
isfactory, and a comparison of condensers of marine engines 
shows a wide diversity of practice. Seaton says that with 
an initial temperature of 6o°, and with 120 for the feed- 
water, a condensation of 13 pounds of steam per square foot 
per hour is considered fair work. A new condenser in good 
condition may condense much more steam per square foot 
per hour than this, but allowance must be made for fouling 
and clogging, especially for vessels that make long voyages. 

Seaton also gives the following table of square feet of 
cooling surface per indicated horse-power: 

Absolute Terminal Pressure, Square Feet 

Pounds per Square Inch. per I. H. P. 

30 3 

20 2.5 

15 • 2.25 

I2£ 2. 

10 1.8 

8 1.6 

6 1.5 



252 THERMODYNAMICS OF THE STEAM-ENGINE. 

For ships stationed in the tropics, allow 20 per cent more; 
for ships which occasionally visit the tropics, allow 10 per cent 
more; for ships constantly in a cold climate, 10 per cent less 
may be allowed. 

Designing Engines. — The only question that is properly 
discussed here is the probable form of the indicator-diagram, 
which gives immediately the method of finding the mean 
effective pressure and, consequently, the size of the cylinder 
of the engine. 

The most reliable way of finding the expected mean 
effective pressure in the design of a new engine is to measure 
an indicator-diagram from an engine of the same or similar 
type and size, and working under the same conditions. 

As it can hardly be expected that a diagram of exactly 
the required form will be at hand, a diagram like Fig. 55 may 
be drawn, using the proper cut-off, com- 
pression, and clearance. If an indicator- 
diagram taken from an engine under similar 
conditions is attainable, it may be used to 

determine exponential equations for the 
Fig. 55. / . „ 

expansion and compression curves; usually 

the exponent will be different for the two curves, and must be 
determined separately. For ordinary work it is sufficient to 
use the hyperbola for both curves, and to assume the steam 
line a and the back-pressure line c to be parallel to the atmos- 
pheric line, while the lead of admission and exhaust may be 
neglected. It is also customary to assume a loss of pressure 
of two or more pounds between the boiler and the engine, 
and a back-pressure of a like amount above the pressure 
in the condenser or the pressure of the atmosphere, as the 
case may be. 

If the diagram is drawn to scale, the area and mean effec- 
tive pressure may be found by measuring it; or, the form of 
the expansion and compression curves being assumed, the 
areas under the steam line, the expansion curve, the back- 
pressure line, and the compression curves may be calculated 



2_ 


.1 











V 6 




\d 






. 2. 


\3 


C 









THE STEAM-ENGINE. 253 

separately, integrating between limits when necessary, and 
therefrom the resulting area of the diagram and the mean 
effective pressure may be determined. Ordinarily, the ex- 
pansion and compression curves are assumed to be hyperbolae. 
Seaton * gives the following multipliers for finding the 
mean effective pressure from that calculated by the process 
described : 

Multipliers for Finding Probable M. E. P., Simple Expansive Engine. 



(1) Special valve-gear, or with separate cut-off valve, 

engine jacketed 

(2) Good ordinary valves, large ports, engine jacketed. 

(3) Ordinary valves and gears as in general practice, 

un jacketed 



0.94 
0.9-0.92 

0.80-0.85 



To estimate the consumption of steam, we may calculate 
from the pressure and volume at release the weight of steam 
then present in the cylinder, and in a similar manner the 
weight of steam caught in the clearance space from the 
volume and pressure at compression, both under the assump- 
tion that the steam is dry and saturated. The difference is 
the steam exhausted per stroke under the assumption; but to 
get a fair estimate of the probable consumption, it is necessary 
to add a fraction of this amount, depending on the style and 
size of the engine and on the conditions under which it is to 
run. Sufficient data for this purpose seldom exist; so it is 
customary to add to the calculated amount one-fourth to one- 
third of itself, to get the probable consumption of non-con- 
densing engines of medium size. 

PROBLEM. — Required the dimensions of an engine to 
give 100 horse-power; revolutions, 120 ; gauge-pressure, 80 
pounds; cut-off at \ stroke; release at end of stroke; com- 
pression at yV stroke, and clearance 5 per cent. 

Assume the pressure during admission to be 78 pounds 
and during exhaust to be 1.3 pounds above the atmosphere, 
and assume hyperbolic expansion and compression. 

* Manual of Marine Engineering. 



254 'THERMODYNAMICS OF THE STEAM-ENGINE. 

The work during expansion is (Fig. 55) 

/i*\ lo & -~= i44(78+i4)(o.333+0-05) P ist - dis P- log. 



0.333+0.05 

so that the mean pressure per square inch during expansion is 

!-05 
0.383 X 927 log, ( ^g^ 

and the mean effective pressure is 

o.333 X 92.7 + 0.383 X 9 2 -7l°g* ^~- - 16 X 0.9 

0.383 

^ 1 o. 1 5 . 

— 0.15 X 16 log e — - = 49.5 pounds. 

If, further, the stroke of the engine is twice the diameter, 

then 

7td 2 2af w w 

— X — X 120 X 2 X 49-5 
412 

100 = — ; 

33000 

.\ <2?= 12.85, .j = 25.70. 

The volume of the cylinder will be 1.93 cubic feet, and the 
terminal pressure will be 33.8 pounds absolute. At 33.8 
pounds the density of steam is 0.08234, and at 16 pounds it 
is 0.04067. The consumption of steam per horse-power per 
hour, on the assumption of dry steam at release and com- 
pression, will be 

(0.08234X1.05—0.04067X0.15)1.93X2X120X60 



100 



22.3pounds. 



If one-third of this quantity be added, then the estimated 
consumption of steam will be 30 pounds per horse-power per 
hour. 

The calculated dimensions are stated in inches and 
hundredths, but in practice the engine would be made I2§ 
inches in diameter by 25 j inches stroke; or possibly the 
dimensions 13 by 25 would be chosen, since they give nearly 
the same volume. 



CHAPTER XII. 



COMPOUND ENGINES. 



A COMPOUND engine has commonly two cylinders, one of 
which is three or four times as large as the other. Steam 
from a boiler is admited to the small cylinder, and after doing 
work in that cylinder it is transferred to the large cylinder, 
from which it is exhausted, after doing work again, into a 
condenser or against the pressure of the atmosphere. If we 
assume that the steam from the small cylinder is exhausted 
into a large receiver, the back-pressure in that cylinder and 
the pressure during the admission to the large cylinder will 
be uniform. If, further, we assume that there is no clearance 
in either cylinder, that the back-pressure in the small cylinder 
and the forward pressure in the large cylinder are the same, 
and that the expansion in the small cylinder reduces the 
pressure down to the back-pressure in that cylinder, the 
diagram for the small cylinder will be ABCD, Fig. 52, and 



r 

"a" 

D 


1B 


sc- 




G 

■0- 




, E 






lr m 










v^k- 



Fig. 56. Fig. 57. 

for the large cylinder DCFG. The volume in the large 
cylinder at cut-off is equal to the total volume of the small 
cylinder, since the large cylinder takes from the receiver the 
same weight of steam that is exhausted by the small cylinder, 
and, in this case, at the same pressure. 

The case just discussed is one extreme. The other 

255 



256 THERMODYNAMICS OF THE STEAM-ENGINE. 

extreme occurs when the small cylinder exhausts directly into 
the large cylinder without an intermediate receiver. In such 
engines the pistons must begin and end their strokes together. 
They may both act on the beam of a beam engine, or they 
may act on one crank or on two cranks opposite each other. 

For such an engine, ABCD> Fig. 57, is the diagram for 
the small cylinder. The steam line and expansion line AB 
and BC are like those of a simple engine. When the piston 
of the small cylinder begins the return stroke, communication 
is opened with the large cylinder, and the steam passes from 
one to the other, and expands to the amount of the difference 
of the volume, it being assumed that the communication 
remains open to the end of the stroke. The back-pressure 
line CD for the small cylinder, and the admission line HI for 
the large cylinder, gradually fall on account of this expansion. 
The diagram for the large cylinder is HIKG, which is turned 
toward the left for convenience. 

To combine the two diagrams, draw the line abed, parallel 
to V'OV, and from b lay off bd equal to ca\ then d is one 
point of the expansion curve of the combined diagram. The 
point C corresponds with H y and E, corresponding with I, is 
as far to the right as / is to the left. 

For a non-conducting cylinder, the combined diagram for 
a compound engine, whether with or without a receiver, is 
the same as that for a simple engine which has a cylinder the 
same size as the large cylinder of the compound engine, and 
which takes at each stroke the same volume of steam as the 
small cylinder, and at the same pressure. The only advan- 
tage gained by the addition of the small cylinder to such an 
engine is a more even distribution of work during the stroke, 
and a smaller initial stress on the crank-pin. 

Compound engines may be divided into two classes — those 
with a receiver and those without a receiver; the latter are 
called " Woolf engines" on the continent of Europe. En- 
gines without a receiver must have the pistons begin and end 
their strokes at the same time; they may act on the same 



COMPOUND ENGINES. 2 $7 

crank or on cranks 180 apart. The pistons of a receiver 
compound engine may make strokes in any order. A form 
of receiver compound engine with two cylinders, commonly 
used in marine work, has the cranks at 90 to give handiness 
and certainty of action. Large marine engines have been 
made with one small cylinder and two large or low-pressure 
cylinders, both of which draw steam from the receiver and 
exhaust to the condenser. Such engines usually have the 
cranks at 120 , though other arrangements have been made. 

Nearly all compound engines have a receiver, or a space 
between the cylinders corresponding to one, and in no case is 
the receiver of sufficient size to entirely prevent fluctuations of 
pressure. In the later marine work the receiver has been 
made small, and frequently the steam-chests and connecting 
pipes have been allowed to fulfil that function. This contrac- 
tion of size involves greater fluctuations of pressure, but for 
other reasons it appears to be favorable to economy. 

Under proper conditions there is a gain from using a com- 
pound engine instead of a simple engine, which may amount 
to ten per cent or more. This gain is to be attributed to the 
division of the change of temperature from that of the steam at 
admission to that of exhaust into two stages, so that there is 
less fluctuation of temperature in a cylinder and consequently 
less interchange of heat between the steam and the walls of 
the cylinder. 

Compound Engine without Receiver. — The indicator- 
diagrams from a compound engine without a receiver are 
represented by Fig. 58. The steam line and 
expansion line of the small cylinder, AB and 
BC, do not differ from those of a simple 
engine. At C the exhaust opens, and the 
steam suddenly expands into the space be- 
tween the cylinders and the clearance of the 
large cylinder, and the pressure falls from 
C to D. During the return stroke the volume in the large 
cylinder increases more rapidly than that of the small cylinder 




258 



THERMODYNAMICS OF THE STEAM-ENGINE. 



decreases, so that the back-pressure line DE gradually falls, 
as does also the admission line HI of the large cylinder, the 
difference between these two lines being due to the resistance 
to -the flow of steam from one to the other. At E the com- 
munication between the two cylinders is closed by the cut-off 
of the large cylinder; the steam is then compressed in the 
small cylinder and the space between the two cylinders to E, 
at which the exhaust of the small cylinder closes; and the 
remainder of the diagram EGA is like that of a simple engine. 
From /, the point of cut-off of the large cylinder, the 
remainder of the diagram IKLMNH is like the same part of 
the diagram of a simple engine. 

The difference between the lines ED and HI and the 
" drop " CD at the end of the stroke of the small piston indi- 
cate waste or losses of efficiency. The compression EFG and 
the accompanying independent expansion IK in the large 
cylinder show a loss of power when compared with a diagram 
like Fig. 56 for an engine which has no clearance or inter- 
mediate space; but compression is required to fill waste 
spaces with steam. The compression EF is required to 
reduce the drop CD, and the compression EG fills the clear- 
ance in anticipation of the next supply from the boiler. 

Neither compression is complete 
in Fig. 58. 

Diagrams from a pumping en- 
gine at Lawrence, Massachusetts, 
are shown by Fig. 59. The 
rounding of corners due to the 
indicator make it difficult to de- 
termine the location of points like 
D, E, E, and I on Fig. 58. The 
low-pressure diagram is taken 
with a weak spring, and so has an 
FlG - 59- exaggerated height. 

Compound Engine with Receiver. — It has already been 
pointed out that some receiver space is required if the cranks 







COMPOUND ENGINES. 259 

of a compound engine are to be placed at right angles. 

When the receiver space is small, as on most marine engines, 

the fluctuations of pressure in the 

receiver are very notable. This is 

exhibited by the diagrams in Fig. 60, 

which were taken from a yacht engine. 

An intelligent conception of the 

causes and meaning of such fluctuations 

can be best obtained by constructing 

ideal diagrams for special cases, as ex- Fig. 60. 

plained on page 269. 

Triple and Quadruple Compound Engines. — The same 
influences which introduced the compound engines, when the 
common steam-pressure changed from forty to eighty pounds 
to the square inch, have brought in the successive expansion 
through three cylinders (the high-pressure, intermediate, and 
low-pressure cylinders) now that 125 to 170 pounds pressure 
are employed. Just as three or more cylinders are combined 
in various ways for compound engines, so four, five, or six 
cylinders have been arranged in various manners for triple- 
expansion engines; for example, a compound engine with 
two cylinders may be conveniently changed into a triple-ex- 
pansion engine by the addition of a small high-pressure cylin- 
der over each of the existing cylinders. 

Quadruple engines with four successive expansions have 
been employed with high-pressure steam, but with the advis- 
able pressures for present use the extra complication and fric- 
tion make it a doubtful expedient. 

Total Expansion. — In Figs. 56 and 57, representing the 
diagrams for compound engines without clearance and without 
drop between the cylinders, the total expansion is the ratio 
of the volumes at E and at A ; that is, of the low-pressure 
piston displacement to the displacement developed by the 
high-pressure piston at cut-off. The same ratio is called the 
total or equivalent expansion for any compound engine, 
though it may have both clearance and drop. Such a con- 



260 THERMODYNAMICS OF THE STEAM-ENGINE. 

ventional total expansion is commonly given for compound 
and multiple-expansion engines, and is a convenience because 
it is roughly equal to the ratio of the initial and terminal 
pressures; that is, of the pressure at cut-ofT in the high- 
pressure cylinder and at release in the low-pressure cylinder. 
It has no real significance, and is not equivalent of the expan- 
sion in the cylinder of a simple engine, by which we mean 
the ratio of the volume at release to that at cut-off, taking 
account of clearance. And further, since the clearance of the 
high- and the low-pressure cylinders are different there can 
be no real equivalent expansion. 

If the ratio of the cylinders is R and the cut-off of the 

high-pressure cylinder is at — of the stroke, then the total 

expansion is represented by 

E = eR ...... (263) 



and 

I 

e 



= R+-E (264) 



This last equation is useful in determining approximately the 
cut-off of the high-pressure cylinder. 

For example, if the initial pressure is 100 pounds absolute 
and the terminal pressure is to be 10 pounds absolute, then 
the total expansions will be about 10. If the ratio of the 
cylinders is 3^, then the high-pressure cut-off will be about 

- = 3 £ -=- IO = O.35 

e 

of the stroke. 

Low-pressure Cut-off. — The cut-off of the low-pressure 
cylinders in Figs. 56 and 57 is controlled by the ratio of the 
cylinders, since the volumes in the low-pressure cylinder at 
cut-off is in each case made equal to the high-pressure piston 
displacement; this is done to avoid a drop. If the cut-off 



COMPOUND ENGINES. 



261 




were lengthened there would be a loss of pressure or drop at 

the end of the stroke of the high-pressure piston, as is shown 

by Fig. 61, for an engine with a 

large receiver and no clearance. 

Such a drop will have some effect on 

the character of the expansion line 

C"E of the low-pressure cylinder, 

both for a non-conducting and for 

the actual engine with or without a 

clearance. Its principal effect will, 

however, be on the distribution of 

work between the cylinders; for it 

is evident that if the cut-off of the low-pressure cylinder is 

shortened the pressure at C" will be increased because the 

same weight of steam is taken in a smaller volume. The 

back-pressure DC of the high-pressure cylinder will be raised 

and the work will be diminished; while the forward pressure 

DC" of the low-pressure cylinder will be raised, increasing the 

work in that cylinder. 

Ratio of Cylinders. — In designing compound engines, 
more especially for marine work, it is deemed important for 
the smooth action of the engine that the total work shall be 
evenly distributed upon the several cranks of the engines, and 
that the maximum pressure on each of the cranks shall be the 
same, and shall not be excessive. In case two or more pis- 
tons act on one crank, the total work and the resultant 
pressure on those pistons are to be considered ; but more 
commonly each piston acts on a separate crank, and then the 
work and pressure on the several pistons are to be considered. 

In practice both the ratio of the cylinders and the total 
expansions are assumed, and then the distribution of work 
and the maximum loads on the crank-pins are calculated, 
allowing for clearance and compression. Designers of engines 
usually have a sufficient number of good examples at hand to 
enable them to assume these data. In default of such data 
it may be necessary to assume proportions, to make prelimi- 



262 THERMODYNAMICS OF THE STEAM-ENGINE. 

nary calculations, and to revise the proportions till satisfactory 
results are obtained. For compound engines using 80 pounds 
steam-pressure the ratio is 1 : 3 or 1 : 4. For triple-ex- 
pansion engines the cylinders may be made to increase in 
the ratio 1 : 2 or I : 2-J. 

Approximate Indicator-diagrams. — The indicator-dia- 
grams from some compound and multiple-expansion engines 
are irregular and apparently erratic, depending on the 
sequence of the cranks, the action of the valves, and the 
relative volumes of the cylinders and the receiver spaces. A 
good idea of the effects and relations of these several influ- 
ences can be obtained by making approximate calculations of 
pressures at the proper parts of the diagrams by a method 
which will now be illustrated. 

For such a calculation it will be assumed that all expan- 
sion lines are rectangular hyperbolas, and in general that any 
change of volume will cause the pressure to change inversely 
as the volume. Further, it will be assumed that when com- 
munication is opened between two volumes where the pres- 
sures are different, the resultant pressure may be calculated 
by adding together the products of each volume by its 
pressure, and dividing by the sum of the volumes. Thus if 
the pressure in a cylinder having the volume v c is p c , and if 
the pressure is p r in a receiver where the volume is v v , then 
after the valve opens communication from the cylinder to the 
receiver the pressure will be 

Pc V c -\-prVr 



P = 



'c I 



V. 



The same method will be used when three volumes are put 
into communication. 

It will be assumed that there are no losses of pressure due 
to throttling or wire-drawing; thus the steam line for the 
high-pressure cylinder will be drawn at the full boiler-pressure, 
and the back-pressure line in the low-pressure cylinder will be 
drawn to correspond with the vacuum in the condenser. 



COMPOUND ENGINES, 



263 



Again, cylinders and receiver spaces in communication will be 
assumed to have the same pressure. 

For sake of simplicity the motions of pistons will be 
assumed to be harmonic. 

Diagrams constructed in this way will never be realized in 
any engine; the degree of discrepancy will depend on the 
type of engine and the speed of rotation. For slow-speed 
pumping-engines the discrepancy is small and all irregularities 
are easily accounted for. On the other hand the discrepancies 
are large for high-speed marine-engines, and for compound 
locomotives they almost prevent the recognition of the events 
of the stroke. 

Direct-expansion Engine. — If the two pistons of a com- 
pound engine move together or in opposite directions the 
diagrams are like those shown by Fig. 62. Steam is admitted 
to the high-pressure cylinder from a to b; cut-off occurs at b, 
and be represents expansion to the end of the stroke; be being 
a rectangular hyperbola referred to the axis (9 J 7 and OP, from 
which a, b, and e are laid off to represent absolute pressures 
and volumes, including clearance. 







p' 



Vr~> 




Fig. 62. 

At the end of the stroke release from the high-pressure 
cylinder and admission to the • low-pressure cylinder are 
assumed to take place, so that communication is opened from 
the high-pressure cylinder through the receiver space into 
the low-pressure cylinder. As a consequence the pressure 
falls from e to d, and rises from n to h in the low-pressure 
cylinder. The line O'P' is drawn at a distance from OP, 



264 THERMODYNAMICS OF THE STEAM-ENGINE. 

which corresponds to the volume of the receiver space, and 
the low-pressure diagram is referred to O'P' and O'V as axes. 

The communication between the cylinders is maintained 
until cut-off occurs at i for the low-pressure cylinder. The 
lines de and hi represent the transfer of steam from the high- 
pressure to the low-pressure cylinder, together with the 
expansion due to the increased size of the large cylinder. 
After the cut-off at i, the large cylinder is shut off from the 
receiver and the steam in it expands to the end of the stroke. 
The back-pressure and compression lines for that cylinder are 
not affected by compounding, and are like those of a simple 
engine. Meanwhile the small piston compresses steam into 
the receiver, as represented by ef, till compression occurs, 
after which compression into the clearance space is represented 
by fg. The expansion and compression lines ik and mn are 
drawn as hyperbolae referred to the axis O'P' and O'V. 
The compression line ef is drawn as an hyperbola referred to 
O'V and O'P', while fg is referred to OV and OP. 

In Fig. 62 the two diagrams are drawn with the same scale 
for volume and pressure, and are placed so as to show to the 
eye the relations of the diagrams to each other. Diagrams 
taken from such an engine resemble Figs. 59, which have the 
same length, and different vertical scales depending on the 
springs used in the indicators. 

Some engines have only one valve to give release and 
compression for the high-pressure cylinder and admission and 
cut-off for the low-pressure cylinder. In such case there is 
no receiver space, and the points e and /"coincide. 

When the receiver is closed by the compression of the 
high-pressure cylinder it is filled with steam with the pressure 
represented by f. It is assumed that the pressure in the 
receiver remains unchanged till the receiver is opened at the 
end of the stroke. It is evident that the drop cd may be 
reduced by shortening the cut-off of the low-pressure cylinder 
so as to give more compression from e to f and consequently 
a higher pressure at /"when the receiver is closed. 



COMPOUND ENGINES. 265 

Representing the pressure and volume at the several points 
by / and v Avith appropriate subscript letters, and represent- 
ing the volume of the receiver by v r , we have the following 
equations: 

p a = p b — initial pressure ; 

p t = p m = back-pressure ; 

pc — pbVb -*- v c ; 

Pn = Pm V m "^ V n \ 

Pd = Ph ~ {pcVc +PnVn +PfVr) -T- tyc + V n + V r ) J 

Pe = Pi = Pd&c + Vn + V r ) -7- (V e -f- V 4 + V r ) ) 

Pf = Av>« + v r ) -r {v f + v r ) ; 

Pg = Pf'f + V £ 5 

p k — P&i -f- v k . 

The pressures/, and p n can be calculated directly. Then 
the equations ior p d , p e , and/ form a set of three simulta- 
neous equations with three unknown quantities, which can be 
easily solved. Finally, p s and p k may be calculated directly. 

For example, let us find the approximate diagram for a 
direct-expansion engine which has the low-pressure piston 
displacement equal to three times the high-pressure piston 
displacement. Assume that the receiver space is half the 
smaller piston displacement, and that the clearance for each 
cylinder is one-tenth of the corresponding piston displace- 
ment. Let the cut-off for each cylinder be at half-stroke, 
and the compression at nine-tenths of the stroke; let the 
admission and release be at the end of the stroke. Let the 
initial pressure be 65.3 pounds by the gauge or 80 pounds 
absolute, and let the back-pressure be two pounds absolute. 

If the volume of the high-pressure piston displacement be 
taken as unity, then the several required volumes are: 

Vb = 0.5 + 0.1 = 0.6 v h -i/„ = 3X 0.1 = 0.3 

v e = v d = 1.0 + 0.1 = 1.1 Vi— 3(0.5 +0.1) = 1.8 

v* =o-5 +Q- 1 = °- 6 »* = »#== 3(i-o + 0.1) = 3.3 

v f = 0.1 + 0.1 = 0.2 v m = 3(0.1 + o. 1) = 0.6 

v s — 0.1 v r — 0.5 



266 THERMODYNAMICS OF THE STEAM-ENGINE. 

The pressures may be calculated as follows: 

Pa — Pb = 80 ; p L —p m = 2 ; 

Pc = pbV b -=- ^ = 8oXo.6-ri.i — 43.6; 

/* = PnPm v^ = 2Xo.6vO,3=:4; 

Pe =PJV C + V n + V r ) -r- (V e + V t + Z> r ) = p d {l.l + O.3 + 0.5) 

~(o.6+ 1.8 +0.5) = 0.655^; 

A = A(^ + »r) "^ 0/ + Vr) = Pe(0.6 + O.5) ~ (0.2 + O.5) 

= 1.57/,= 1.57 X 0.655/,= 1.03^; 

A = (PcV c +PnV n +PfV r ) -T- (^ c + Z/„ + Z> r ) 

= (80 X 0.6 + 4 Xo.3+0.5^) -^0.6 + 0.3+0.5 

= 25.89+0.26^; 
Pd = 25.89 + 0.26 X 1.03/, ; p d = 35.36 ; 
p e = Pi = o.6$$p d = 0.655 X 35-36 = 23.2 ; 
P f = I-Q3A = ^ X 35-36 = 36.5 ; 
Pe =Pf v f --' v e = 36.5 X 0.2 ~ 0.1 = 73 ; 
/* =piVi + v k =z 23.2 X 1.8 ~ 3.3 = 12.6. 

As the pressures and volumes are now known the diagrams 
of Fig. 62 may be drawn to scale. Or, if preferred, diagrams 
like Fig. 59 may be drawn, making them of the same length 
and using convenient vertical scales of pressure. If the 
engine runs slowly and has abundant valves and passages the 
diagrams thus obtained will be very nearly like those taken 
from the engine by indicators. If losses of pressure in valves 
and passages are allowed for, a closer approximation can be 
made. 

The mean effective pressures of the diagrams may be 
readily obtained by the aid of planimeters, and may be used 
for estimating the power of the engine. For this purpose we 
should either use the modified diagrams allowing for losses of 
pressure, or we should affect the mean effective pressures by 
a multiplier obtained by comparison of the approximate with 
the actual diagrams from engines of the same type. For a 
slow-speed pumping-engine the multiplier may be as lar^e as 
0.9 or even more; for high-speed engines it may be as small 
as 0.6. 



COMPOUND ENGINES. 267 

The mean effective pressure of the diagrams may be cal- 
culated from the volumes and pressures if desired, assuming, 
of course, that the several expansion and compression curves 
are hyperbolae. The process can be best explained by apply- 
ing it to the example already considered. Begin by finding 
the mean pressure during the transfer of steam from the high- 
pressure cylinder to the low-pressure cylinder as represented 
by de and hi. The net effective work during the transfer is 

pdv = p x v x log, - 2 = i44Mv<i + Vh + v r ) log 2 



= J 44 X 35.4(1.1+0.3 + 0.5) log. 



V d + V h + V, 

0.6 + 1.8 + 0.5 



1.1 +0.3+0.5 
= 4120 foot-pounds" 

for each cubic foot of displacement of the high-pressure piston. 
This corresponds with our previous assumption of unity for 
the displacement of that piston. The increase of volume is 

v e +v i +v r -{v d +^+^=0.6+ 1.8+0.5— (1.1+0.3+0.5)= 1 ; 

so that the mean pressure during the transfer is 

4120 -J- I X 144 = 28.6 — p x 

pounds per square inch, which acts on both the high- and the 
low-pressure pistons. 

The effective work for the small cylinder is obtained by 
adding the works under ab and be and subtracting the works 
under de, ef, and fg. So that 

Wh = 144 i Pa(v — V a ) + pbVb loge px{vd — V c ) 

( Vb 

V e -\- V r Vf ) 

— pe{V e + V r ) log e ■ p Vf \og e — V 

Vf -j- v r v g ) 

= 144^ 80(0.6 — 0.1) -4- 80 X 0.6 log, — — 28.6(1.1 — 0.6) 
( 0.6 

O.6 + O.5 0.2) 

— 23.2(0.6 + 0.5) log, — — ■ 36.5 X 0.2 log, — [ 

0.2 + 0.5 O. I ) 

= 144 X 33-26 = 47S9 foot-pounds. 



268 THERMODYNAMICS OF THE STEAM-ENGINE. 

This is the work for each cubic foot of the high-pressure 
piston displacement, and the mean effective pressure on the 
small piston is evidently 33.26 pounds per square inch. 

In a like manner the work of the large piston is 

Wl — 1441 px{vi — vh) -\- pivi loge pi{yi — v m ) — p m v m log* — J 

( Vi Vn ) 

= 144 J 28.6(1.8 - 0.3) + 23.2 X 1.8 log, ^|- 

— 2(3.3 — 0.6)— 2 X 0.6 loge — t = 144 X 61.92 = 8916 foot-pounds. 



Since the ratio of the piston displacements is 3, the work 
for each cubic foot of the low-pressure piston displacement is 
one-third of the work just calculated; and the mean effective 
pressure on the large piston is 

61.92 -^3 = 20.64 

pounds per square inch. 

The proportions given in the example lead to a very 
uneven distribution of work; that of the large cylinder being 
nearly twice as much as is developed in the small cylinder. 
The distribution can be improved by shortening the cut-off 
of the small cylinder, or lengthening that of the large 
cylinder, or by increasing the size of the large cylinder. 

As has already been pointed out, the works just calculated 
and the corresponding mean effective pressures are in excess 
of those that will be actually developed, and must be affected 
by multipliers which may vary from 0.6 to 0.9, depending on 
the type and speed of the engine. 

Cross-compound Engine. — A two-cylinder compound 
engine with pistons connected to cranks at right angles with 
each other is frequently called a cross-compound engine. 
Unless a large receiver is placed between the cylinders the 
pressure in the space between the cylinders will vary widely. 

Two cases arise in the discussion of this engine according 
as the cut-off of the large cylinder is earlier or later than half- 



COMPOUND ENGINES. 



269 



stroke ; in the latter case there is a species of double admission 
to the low-pressure cylinder, as is shown in Fig. 63. For 
sake of simplicity the release, and also the admission for each 
cylinder, is assumed to be at the end of the stroke. If the 
release is early the double admission occurs before half-stroke. 
The admission and expansion of steam for the high- 
pressure cylinder are represented by ab and be. At c release 
occurs, putting the small cylinder in communication with the 
intermediate receiver, which is then open to the large cylinder. 




Fig. 03. 

There is a drop at cd and a corresponding rise of pressure mn 
on the large piston, which is then at half-stroke; it will be 
assumed that the pressures at d and at n are identical. From 
d to e the steam is transferred from the small to the large 
'cylinder, and the pressure falls because the volume increases; 
no is the corresponding line on the low-pressure diagram. 
The cut-off at for the large cylinder interrupts this transfer, 
and steam is then compressed by the small piston into the 
intermediate receiver with a rise of pressure as represented 
by ef. The admission to the large cylinder, tk, occurs when 
the small piston is at the middle of its stroke, and causes a 
drop, fg t in the small cylinder. From g to h steam is trans- 
ferred through the receiver from the small to the large 
cylinder. The pressure rises at first because the small piston 
moves rapidly while the large one moves slowly until its crank 
gets away from the dead-point ; afterwards the pressure falls. 
The line £/ represents this action on the low-pressure diagram. 
At h compression occurs for the small cylinder, and hi shows 



270 THERMODYNAMICS OF THE STEAM-ENGINE. 

the rise of pressure due to compression. For the large 
cylinder the line Im represents the supply of steam from the 
receiver, with falling pressure which lasts till the double 
admission at mn occurs. 

The expansion, release, exhaust, and compression in the 
large cylinder are not affected by compounding. 

Strictly, the two parts of the diagram for the low-pressure 
cylinder, mnopq and stklm belong to opposite ends of the 
cylinder, one belonging to the head end and one to the crank 
end. With harmonic motion the diagrams from the two ends 
are identical, and no confusion need arise from our neglect to 
determine which end of the large cylinder we are dealing with 
at any time. Such an analysis for the two ends of the 
cylinder, taking account of the irregularity due to the action 
of the connecting-rod, would lead to a complexity that would 
be unprofitable. 

A ready way of finding corresponding positions of two 
pistons connected to cranks at right angles with each other is 

by aid of the diagram of Fig. 60. Let 
O be the centre of the crank-shaft and 
pR y R x q be the path of the crank-pin. 
When one piston has the displacement py 
and its crank is at 0R y , the other crank* 
may be 90 ahead at 0R X and the corre- 
tT "T sponding piston displacement will be px. 

The same construction may be used if 
the crank is 90 behind or if the angle R y OR x is other than a 
right angle. The actual piston position and crank angles 
when affected by the irregularity due to the connecting-rod 
will differ from those found by this method, but the position 
found by such a diagram will represent the average positions 
very nearly. 

The several pressures may be found as follows: 

Pb = pa — initial pressure ; 
p s = p g = back-pressure ; 

Pc — pbV b -T- v e ; 




COMPOUND ENGINES. 2*]\ 

Pt — P*V S 4- v t ; 

Pd=Pn= \pcV c +Pm(v m +V r )\ -f" (V e + V m -f Z^) J 
Pe—Pa= Pd(?c + #* + «V) -5- (^ + ^ + «V) I 
Pf = Pe(v e + Vr) + {Vf + V r ) J 

A = A = f P/(v f + «v) +p t v t ] ~ (v f + ^ + v r ) ; 

A = A =PA V /+ V t + v r ) ~- z/ A + ^ + v r ) ; 

A* = Afe + «V) -T- (^* + ^r) ; 

A = A^a rr- ^ ; 
A = A^ -r- v p . 

The pressures / c and p n can be found directly from the 
initial pressure and the back-pressure, and finally the last two 
equations give direct calculations for p t and p p so soon as p h 
and p are found. There remain six equations containing six 
unknown quantities, which can be readily solved after numeri- 
cal values are assigned to the known pressures and to all the 
volumes. 

The expansion and compression . lines, be and hi, for the 
high-pressure diagrams are hyperbolae referred to the axis OV 
and OP; and in like manner the expansion and compression 
lines op and st, for the low-pressure diagram are hyperbolae 
referred to O' V and O' P' . The curve ef is an hyperbola 
referred to O'V and O'P', and the curve Im is an hyperbola 
referred to OV and OP. The transfer lines de and no, gJi 
and kl, are not hyperbolae. They may be plotted point by 
point by finding corresponding intermediate piston positions, 
A an d p y t by aid of Fig. 64, and then calculating the pressure 
for these positions by the equation 

Px—Py — Pd(v d + V m + V r ) -r- (v x + V y + V r ). 

The work and mean effective pressure may be calculated 
in a manner similar to that given for the direct-expansion 
engine; but the calculation is tedious, and involves two trans- 
fers, de and no, and gh and kl. The first involves only an 
expansion, and presents no special difficulty ; the second con- 
sists of a compression and an expansion, which can be dealt 



272 THERMODYNAMICS OF THE STEAM-ENGINE. 

with most conveniently by a graphical construction. All 
things considered, it is better to plot the diagrams to scale 
and determine the areas and mean effective pressures by aid 
of a planimeter. 

If the cut-off of the low-pressure is earlier than half-stroke 
so as to precede the release of the high-pressure cylinder the 
transfer represented by de and no, Fig. 63, does not occur. 
Instead there is a compression from d to /"and an expansion 
from / to m. The number of unknown quantities and the 
number of equations are reduced. On the other hand a 
release before the end of the stroke of the high-pressure piston 
requires an additional unknown quantity and one more 
equation. 

Triple Engines. — The diagrams from triple and other 
multiple-expansion engines are likely to show much irregu- 
larity, the form depending on the number 
and arrangement of the cylinders and the 
sequence of the cranks. A common ar- 
rangement for a triple engine is to have 
three pistons acting on cranks set equidis- 
tant around the circle, as shown by Fig. 
65. Two cases arise depending on the 
sequence of the cranks, which may be in the 
order, from one end of the engine, of high-pressure, low- 
pressure, and intermediate, as shown by Fig. 65 ; or in the 
order of high-pressure, intermediate, and low-pressure. 

With the cranks in the order high-pressure, low-pressure, 
and intermediate, as shown by Fig. 65, the diagrams are like 
those given by Fig. 66. The admission and expansion for 
the high-pressure cylinder are represented by abc. When the 
high-pressure piston is at release, its crank is at H y Fig. 65, 
and the intermediate crank is at /, so that the intermediate 
piston is near half-stroke. If the cut-off for that cylinder is 
later than half-stroke, it is in communication with the first 
receiver when its crank is at /, and steam may pass through 
the first receiver from the high-pressure to the intermediate 




COMPOUND ENGINES. 



273 



cylinder, and there is a drop cd, and a corresponding rise of 
pressure no in the intermediate cylinder. The transfer con- 
tinues till cut-off for the intermediate cylinder occurs at /, 
corresponding to the piston position e for the high-pressure 




y 




z 


ft 

a 


r 

—~ — „__ 
1 
1 
1 


~- 6 






1 

Atmospheric line 


1 

1 

-j 

I 
1 
1 




V 


^ .c 


1 
1 

Scale 40 


€ 













Fig. 66. 
cylinder. From the position e the high-pressure piston moves 
to the end of the stroke at/, causing an expansion, and then 
starts to return, causing the compression fg. When the 
high-pressure piston is at g the intermediate cylinder takes 
steam at the other end causing the drop gh and the 
rise of pressure xl. Then follows a transfer of steam 
from the high-pressure to the intermediate cylinder repre- 
sented by hi and Im. At i the high-pressure compression ik 



274 THERMODYNAMICS OF THE STEAM-ENGINE. 

begins, and is carried so far as to produce a loop at k. After 
compression for the high-pressure cylinder shuts it from the 
first receiver, the steam in that receiver and in the inter- 
mediate cylinder expand as shown by mn. The expansion in 
the intermediate cylinder is represented by pq and the release 
by qr, corresponding to a rise of pressure a/3 in the low- 
pressure cylinder, rs and j3y represent a transfer of steam 
from the intermediate cylinder to the low-pressure cylinder. 
The remainder of the back-pressure line of the intermediate 
cylinder and the upper part of the low-pressure diagram for 
the low-pressure cylinder correspond to the same parts of the 
high-pressure and the intermediate cylinders, so that a state- 
ment of the actions in detail does not appear necessary. 

The equations for calculating the pressure are numerous, 
but they are not difficult to state, and the solution for a given 
example presents no special difficulty. Thus we have 

p a = pb = initial pressure; 
pc = pbVb -*- v c ; 

pd— po — { pcVc + Pn{v + V r ) ) ■+■ (v d + V + V r ) \ 

pe—pp= pdiyd + V + Vr) -f- V e + Vp -f V r )\ 

Pf — Pe{Ve + V r ) "*- (Vf + V r )\ 

PS = Pf( v f + *r) -S- (Vg + »r)j 

ph = Pi = { Pg(Vg- + »r) + /*W* } -5- (v* + VJ + ^); 

pi = pm = ph(Vh + »/ 4" »r) + (V» +f« + Vr); 

/* = A'Vt -s- v*; 

/n = /*(*« + V r ) "*- (»» + V r ); 

Ps = Ppvp -*- v,?; 

/r = //3 = 1 /V<? -f- ^a(>a -f- W/?;J -*■ (l/ r + V a + Vff); 
p s — p y — p r {y r + Va + Vi?) -S- (w s + V-y -f- Z/#); 

/z = /> s (v s + v/?) -f- (^ + v/?); 
/« = /*(v* + v/?) -*- (»i» + vi?) ; 

A> = { Pu{Vu 4~ V7?) + p-qV-q } -*- (v v + Vy, -\- V r)\ 

pw= pv{vv -f Vr, 4" Vi?) -5" (Z/ W 4" V z 4~ V/?) J 

/* = ^7«V W -J- Z/jcI 

pa = (Vz 4- Vtf) -4- (z/ a + V^)j 

^S = paVa -T- ^Si 

p e = p$ = back-pressure; 

Pr, = piVi, -5- 7/r,. 



COMPOUND ENGINES. 2/5 

The pressures at c and at r/ can be calculated immediately 
from the initial pressure and from the back-pressure. Then it 
will be recognized that there are four individual equations for 
finding^, p k1 p t , and/ 5 . The fourteen remaining equations, 
solved as simultaneous equations, give the corresponding 
fourteen required pressures, some of which are used in calcu- 
lating the four pressures which are determined by the four 
individual equations. 

If the cut-off for the intermediate cylinder occurs before 
the release of the high-pressure cylinder, then the transfer 
represented by de and op does not occur. In like manner, if 
the cut-off for the low-pressure cylinder occurs before the 
release for the intermediate cylinder, the transfer represented 
by rs and. fiy does not occur. The omission of a transfer of 
course simplifies the work of deducing and of solving equa- 
tions. 

In much the same way, equations may be deduced for 
calculating pressures when the cranks have the sequence 
high pressure, intermediate, and low-pressure. The dia- 
grams take forms which are distinctly unlike those for the 
other sequence of cranks. In general, the case solved, 
i.e., high-pressure, low-pressure, and intermediate, gives a 
smoother action for the engine. 

For example, the engines of the U. S. S. Machias have 
the following dimensions and proportions: 

High- Inter- Low- 

pressure, mediate, pressure. 

Diameter of piston, inches I5f 22^ 35 

Piston displacement, cubic feet 2.71 5-53 13-39 

Clearance, per cent 13 14 7 

Cut-off, per cent stroke 66 66 66 

Release, " " 93 93 93 

Compression, per cent stroke 18 18 18 

Ratio of piston displacements I 2.04 4.94 

Volume first receiver, cubic feet 2.22 

Volume second " " 6.26 

Ratio of receiver volumes to high-pressure piston 

displacement 0.82 2.31 

Stroke, inches 24 

Boiler-pressure, absolute, pounds per sq. in 180 

Pressure in condenser, " «•<<<< 2 



276 



THERMODYNAMICS OF THE STEAM-ENGINE. 



If the volume of the high-pressure piston displacement is 
taken to be unity, then the volumes required in the equations 






Fig. 67. 

for calculating pressures, when the cranks have the order high- 
pressure, low-pressure, and intermediate, are as follows: 



v b =0.79 

Vc •— V d = 1 .06 
V e — 1. 10 

v f = 1. 13 
v s =v h — 0.88 
v { = 0.31 
v k = v a = 0.13 



v t = v x = 0.29 

v m — 0.98 

v n — v — 1.26 

Vp = 1.63 

v q = v r = 2.18 

z/ s = 2.28 

^ = 2.33 

v u = v v = 1.85 
v m = o 63 



^ = ^ = °-35 
z/ z — 2.02 

v* = ^ = 2.72 
z/ v = 3.60 
z;a = z/ € = 4.94 
v i = 1.23 






COMPOUND ENGINES. 2-Jf 

The required pressures are: 

A=/* =I 5o A=i6s /«=/, = 25-6 

A = 112 / tt = 6o.o A =52-3 

A=A=7 6 -5 A =50-5 A = 22.1 

A=A = 6 7-5 A=A = 28.3 /« = 18.5 

A = 6 7-5 A = A = 25.3 A = A = 5 

A = 7 6 -9 A =25.1 >„=i7.6 

A=A = 73-5 A =29.0 

A =Pm = 69.3 A = A = 28.2 

Diagrams with the volumes and pressures corresponding 
to this example are plotted in Fig. 66 with convenient 
vertical scales. Actual indicator-diagrams taken from the 
engine are given by Fig. 67. The events of the stroke come 
at slightly different piston positions on account of the irregu- 
larity due to the connecting-rod, and the drops and other 
fluctuations of pressure are gradual instead of sudden; more- 
over, there is considerable loss of pressure from the boiler to 
the engine, from one cylinder to another, and from the low- 
pressure cylinder to the condenser. Nevertheless the general 
forms of the diagrams are easily recognized and all apparent 
erratic variations are accounted for. 

Designing Compound Engines. — The designer of com- 
pound and multiple-expansion engines should have at hand 
a well-systematized fund of information concerning the sizes, 
proportions, and powers of such engines, together with actual 
indicator-diagrams. He may then, by a more or less direct 
method of interpolation or exterpolation, assign the dimen- 
sions and proportions to a new design, and can, if deemed 
advisable, draw or determine a set of probable indicator- 
diagrams for the new engines. If the new design differs much 
from the engines for which he has information he may deter- 
mine approximate diagrams both for an actual engine from 
which indicator-diagrams are at hand, and for the new design. 
He may then sketch diagrams for the new engine, using the 
approximate diagrams as a basis and taking a comparison of 



278 



THERMODYNAMICS OF THE STEAM-ENGINE. 



the approximate and actual diagrams from the engine already 
built, as a guide. It is convenient to prepare and use a table 
showing the ratios of actual mean effective pressures and 
approximate mean effective pressures. Such a table enables 
the designer to assign mean effective pressures to a cylinder 
of the new engine and to infer very closely what its horse- 
power will be. It is also very useful as a check in sketching 
probable diagrams for a new engine, which diagrams are 
not only useful in estimating the power of the new engine, 
but in calculating stresses on the members of that engine. 

A rough approximation of the power of an engine may be 
made by calculating the power as though the total or equiva- 
lent expansion took place in the low-pressure cylinder, neg- 
lecting clearance and compression. The power thus found is 
to be affected by a factor which according to the size and type 
of the engine may vary from 0.6 to 0.9 for compound engines 
and from 0.5 to 0.8 for triple engines. Seaton and Roun- 
thwaite * give the following table of multipliers for compound 
marine engines: 

MULTIPLIERS FOR FINDING PROBABLE M.E.P. COMPOUND 
AND TRIPLE MARINE ENGINES. 



Description of Engine. 

Receiver-compound, screw engines. ........ 

do paddle engines 

Direct expansion 

Three cylinder triple, merchant ships 

do naval vessels 

do gunboats and torpedo 
boats 



Jacketed. 



O.67 to O.73 



O.64 to O.68 
O.55 to 0.65 



Unjacketed. 



O.58 to 068 
O.55 to O.65 
O.71 to 0.75 
O.60 to 0.66 



0.60 to 0.67 



For example, let the boiler-pressure be 80 pounds by the 
gauge, or 94.7 pounds absolute; let the back-pressure be 
4 pounds absolute; and let the total number of expansions be 
six, so that the volume of steam exhausted to the condenser 
is six times the volume admitted from the boiler. Neglect- 

* Pocket Book of Marine Engineering. 






COMPOUND ENGINES. 2?$ 

ing the effect of clearance and compression, the mean effective 
pressure is 

94.7 X |+ 94-7 X i log, f-4X 1 = 40.06 = M.E.P. 

If the large cylinder is 30 inches in diameter, and the 
stroke is 4 feet, the horse-power at 60 revolutions per minute 
is 

tt^o 2 

-=— X 40.06 X 2 X 4 X 60 -r 33000 = 412 H.P. 
4 

If the factor to be used in this case is 0.75, then the actual 
horse-power will be about 

0,75 X 400 = 3 00 H.P. 



CHAPTER XIII. 
TESTING STEAM-ENGINES. 

THE principal objects of tests of steam-engines is to de- 
termine the cost of power. Series of engine tests are made 
to determine the effect of certain conditions, such as super- 
heating and steam-jackets, on the economy of the engine. 
Again, tests may be made to investigate the interchanges of 
heat between the steam and the walls of the cylinder by the 
aid of Hirn's analysis, and thus find how certain conditions 
produce effects that are favorable or unfavorable to economy. 

The two main elements of an engine test are, then, the 
measurement of the power developed and the determination 
of the cost of the power in terms of thermal units, pounds of 
steam, or pounds of coal. Power is most commonly measured 
by aid of the steam-engine indicator, but the power delivered 
by the engine is sometimes determined by a dynamometer or a 
friction break; sometimes, when an indicator cannot be used 
conveniently, the dynamic or break power only is deter- 
mined. When the engine drives a dynamo-electric generator 
the power applied to the generator may be determined elec- 
trically, and thus the power delivered by the engine may be 
known. 

When the cost of power is given in terms of coal per 
horse-power per hour, it is sufficient to weigh the coal for a 
definite period of time. In such case a combined boiler and 
engine test is made, and all the methods and precautions for a 
careful boiler test must be observed. The time required for 

such a test depends on the depth of the fire on the grate and 

280 






TESTING STEAM-ENGINES. 28 1 

the rate of combustion. For factory boilers the test should 
be twenty-four hours long if exact results are desired. 

When the cost of power is stated in terms of steam per 
horse-power per hour, one of two methods may be followed. 
When the engine has a surface condenser the steam exhausted 
from the engine is condensed, collected, and weighed. One 
hour is usually sufficient for tests under favorable conditions; 
shorter intervals, ten or fifteen minutes, give fairly uniform 
results. The chief objection to this method is that the ar 
pumped from the condenser is saturated with moisture which is 
not accounted for. The error from this source is probably not 
important, for the results of tests by this method and by de- 
termining the feed-water supplied to the boiler are substan- 
tially the same. In tests on non-condensing and on jet-con- 
densing engines the steam consumption is determined by 
weighing or measuring the feed-water supplied to the boiler 
or boilers that furnish steam to the engine. Steam used for 
any other purpose than running the engine, for example, for 
pumping, heating, or making determinations of the quality of 
the steam, must be determined and allowed for. The most 
satisfactory way is to condense and weigh such steam, but 
small quantities, as for determining quality by a steam calo- 
rimeter, may be gauged by allowing it to flow through an 
orifice. Tests which depend on measuring the feed-water 
should be long enough to minimize the effect of the uncer- 
tainty of the amount of water in a boiler corresponding to an 
apparent height of water in a water-gauge ; for the apparent 
height of the water-level depends largely on the rate of vapor- 
ization and the activity of convection currents. 

When the cost of power is expressed in thermal units it is 
necessary to measure the steam pressure, the amount of moisture 
in the steam supplied to the cylinder, and the temperature of 
the condensed steam when it leaves the condenser. If steam 
is used in jackets or reheaters it must be accounted for sepa- 
rately. But it is customary in all engine tests to take pressures 
and temperatures, so that the cost may usually be calculated in 



282 THERMODYNAMICS OF THE STEAM-ENGINE. 

thermal units, even when the experimenter is content to state 
it in pounds of steam. 

For a Hirn's analysis it is necessary to weigh or measure 
the condensing water, and to determine the temperatures at 
admission to and exit from the condenser. 

Important engines, with their boilers and other appurte- 
nances, are commonly built under contract to give a certain 
economy, and the fulfilment of the terms of a contract is de- 
termined by a test of the engine or of the whole plant. The 
test of the entire plant has the advantage that it gives, as one 
result, the cost of power directly in coal ; but as the engine is 
often watched with more care during a test than in regular 
service, and as auxiliary duties, such as heating and banking 
fires, are frequently omitted from such an economy test, the 
actual cost of power can be more justly obtained from a rec- 
ord of the engine in regular service, extending for weeks or 
months. The tests of engine and boilers, though made at the 
same time, need not start and stop at the same time, though 
there is a satisfaction in making them strictly simultaneous. 
This requires that the tests shall be long enough to avoid 
drawing the fires at beginning and end of the boiler test. 

In anticipation of a test on an engine it must be run for 
some time under the conditions of the test, to eliminate 
the effects of starting or of changes. Thus engines with 
steam-jackets are commonly started with steam in the jackets, 
even if they are to run with steam excluded from the jackets. 
An hour or more must then be allowed before the effect 
of using steam in the jackets will entirely pass away. An 
engine without steam-jackets, or with steam in the jackets, 
may come to constant conditions in a fraction of that time, but 
it is usually well to allow at least an hour before starting 
the test. 

It is of the first importance that all the conditions of a test 
shall remain constant throughout the tests. Changes of steam- 
pressure are particularly bad, for when the steam-pressure 
rises the temperature of the engine and all parts affected 



TESTING STEAM-ENGINES. 283 

by the steam must be increased, and the heat required for this 
purpose is charged against the performance of the engine ; if 
the steam-pressure falls a contrary effect will be felt. In a series 
of tests one element at a time should be changed, so that the 
effect of that element may not be confused by other changes, 
even though such changes have a relatively small effect. It 
is, however, of more importance that steam-pressure should 
remain constant than that all tests at a given pressure should 
have identically the same steam-pressure, because the total 
heat of steam varies more slowly than the temperature. 

All the instruments and apparatus used for an engine test 
should be tested and standardized either just before or just 
after the test ; preferably before, to avoid annoyance when 
any instrument fails during the test and is replaced by another. 

Thermometers. — Temperatures are commonly measured 
by aid of mercurial thermometers, of which three grades may 
be distinguished. For work resembling that done by the 
physicist the highest grade should be used, and these must 
ordinarily be calibrated, and have their boiling- and freezing- 
points determined by the experimenter or some qualified 
person ; since the freezing-point is liable to change, it should 
be redetermined when necessary. For important data good 
thermometers must be used, such as are sold by reliable 
dealers, but it is preferable that they should be calibrated or 
else compared with a thermometer that is known to be reliable. 
For secondary data or for those requiring little accuracy 
common thermometers with the graduation on the stem may 
be used, but these also should have their errors determined 
and allowed for. Thermometers with detachable scales should 
be used only for crude work. 

Gauges. — Pressures are commonly measured by Bourdon 
gauges, and if recently compared with a correct mercury 
column these are sufficient for engineering work. The columns 
used by gauge-makers are commonly subject to minor errors, 
and are not usually corrected for temperature. It is important 
that such gauges should be frequently retested. From their 



284 THERMODYNAMICS OF THE STEAM-ENGINE. 

convenience vacuum-gauges of the same form are used, even 
where a mercurial gauge could easily be applied. 

The pressure of the atmosphere may be taken with either a 
mercurial or an aneroid barometer, but if the latter is used its 
errors must be known. It should be easy to make the baro- 
metric errors only a fraction of the unavoidable gauge errors. 

Dynamometers. — The standard for measurement of power 
is the friction-brake. For smooth continuous running it is 
essential that the brake and its band shall be cooled by a stream 
of water that does not come in contact with the rubbing surfaces. 
Sometimes the wheel is cooled by a stream of water circulating 
through it, sometimes the band is so cooled, or both may be. 
A rubbing surface which is not cooled should be of non-con- 
ducting material. If both rubbing surfaces are metallic they 
must be freely lubricated with oil. An iron wheel running in 
a band furnished with blocks of wood requires little or no 
lubrication. 

To avoid the increase of friction on the brake-bearings due 
to the load applied at a single brake-arm two equal arms may 
be used with two equal and opposite forces applied at the ends 
to form a statical couple. 

With care and good workmanship a friction-brake may be 
made an instrument of precision sufficient for physical investi- 
gations, but with ordinary care and workmanship it will give 
results of sufficient accuracy for engineering work. 

All forms of transmission dynamometers should be stand- 
ardized, and should have their errors determined by compari- 
son with a friction-brake. 

Indicators. — The most important and at the same time the 
least satisfactory instrument used in engine-testing is the 
indicator. Even when well made and in good condition it is 
liable to have an error of two per cent or more when used at 
moderate speeds. At high speeds, three hundred revolutions 
per minute and over, it is likely to have two or three times as 
much error. As a rule, an indicator cannot be used at more 
than four hundred revolutions per minute. 



TESTING STEAM-ENGINES. 285 

The mechanism for reducing the motion of the crosshead 
of the engine and transferring it to the paper drum of an 
indicator should be correct in design and free from undue 
looseness. It should require only a very short cord leading to 
the paper drum, because all the errors due to the paper drum 
are proportional to the length of the cord and may be prac- 
tically eliminated by making the cord short. 

The weighing and recording of the steam-pressure by the 
indicator-piston, pencil-motion, and pencil are affected by 
errors which may be classified as follows : 

1. Scale of the spring. 

2. Design of the pencil-motion. 

3. Inertia of moving parts. 

4. Friction and backlash. 

Good indicator-springs, when tested by direct loads out of 
the indicator, usually have correct and uniform scales ; that is, 
they collapse under pressure the proper amount for each load 
applied. The springs are perceptibly weaker at high tempera- 
tures than at ordinary temperatures, but when used on a steam- 
engine they are exposed to steam but little, if any, above the 
pressure of the atmosphere, and are but little affected thereby. 
Some recent indicators have the spring above the cylinder, so 
that it is not exposed to steam. If the scale of a spring 
is uniform, but is either more or less than the rated scale, a 
correction can easily be applied. Thus a spring marked for 
50 pounds to the inch may be really a 48-pound spring, if the 
indicator-pencil rises an inch for 48-pounds increase of pres- 
sure. But a spring having an irregular scale must be rejected, 
as there is no convenient way of applying corrections for 
irregular errors, especially when the area of the diagram 
is measured by a planimeter. 

The motion of the piston of the indicator is multiplied five 
or six times by the pencil-motion, which is usually an approx- 
imate parallel motion. Within the proper range of motion 
(about two inches) the pencil draws a line which is nearly 



286 THERMODYNAMICS OF THE STEAM-ENGINE. 

straight when the paper drum is at rest, and it gives a nearly 
uniform scale provided that the spring is uniform. The errors 
due to the geometric design of this part of the indicator are 
always small. 

When steam is suddenly let into the indicator, as at ad- 
mission to the engine-cylinder, the indicator-piston and at- 
tached parts forming the pencil-motion are set into vibration, 
with a natural time of vibration depending on the stiffness of 
the spring. A weak spring used for indicating a high-speed 
engine may throw the diagram into confusion, because the 
pencil will give a few deep undulations which make it impos- 
sible to recognize the events of the stroke of the engine, such 
as cut-off and release. A stiffer spring will give more rapid 
and less extensive undulations, which will be much less trouble- 
some. As a rule, when the undulations do not confuse the 
diagram the area of the diagram is but little affected by the 
undulations due to the inertia of the piston and pencil-motion. 

The largest and most troublesome errors of the indicator 
are due to friction and backlash. The various joints at the 
piston and in the pencil-motion are made as tight as can be 
without undue friction, but there is always some looseness and 
some friction at those joints. There is also some friction of 
the piston in the cylinder and of the pencil on the paper. 
Errors from this source are commonly one or two per cent, 
and may be excessive unless the instrument is used with care 
and skill. A blunt pencil pressed up hard on the paper will 
reduce the area of the diagram five per cent or more; on the 
other hand a medium pencil drawing a faint but legible line 
will affect the area very little. Any considerable friction of 
the piston of the indicator will destroy the value of the diagram. 

Errors of the scale of the spring can be readily determined 
and investigated by loading the spring with known weights, 
when properly supported, out of the indicator. Other tests of 
the indicator are difficult and are likely to be misleading, as in 
general such tests consist in comparing the indicator with either 
a mercury column or a gauge when subjected to the same 



TESTING STEAM-ENGINES. 2'6'J 

pressure. Now a gauge, and still more a mercury column, can 
be used only for measuring steady pressure, while an indicator- 
piston is sure to show excessive friction if it does not stick 
fast when exposed to a steady pressure. The consequence is 
that a test made by changing pressure progressively on both 
an indicator and a gauge or mercury column is likely to lead to 
errors in the indications of both instruments. The only satis- 
factory way of testing an indicator is to subject it to changing 
pressure, so as to simulate the action of steam in the cylinder 
of an engine, and to measure the pressure with a mercury col- 
umn or a gauge. The two pressures may differ by five or ten 
pounds and may be varied progressively at a rate adapted to 
the use of a mercury column or gauge. Very few instruments 
for thus testing indicators have been made, and they have not 
yet been adapted to commercial work. Such investigations 
show that the error of an indicator need not be more than one 
or two per cent. 

Scales. — Weighing should be done on scales adapted to 
the load ; overloading leads to excessive friction at the knife- 
edges and to lack of delicacy. Good commercial platform 
scales, when tested with standard weights, are sufficient for 
engineering work. 

Coal and ashes are readily weighed in barrows, for which 
the tare is determined by weighing empty. Water is weighed 
in barrels or tanks. The water can usually be pumped in or 
allowed to run in under a head, so that the barrel or tank can 
be filled promptly. Large quick-opening valves must be used 
to allow the water to run out quickly. As the receptacle will 
seldom drain properly, it is not well to wait for it to drain, but 
to close the exit-valve and weigh empty with whatever small 
amount of water may be caught in it. Neither is it well to try 
to fill the receptacle to a given weight, as the jet of water 
running in may affect the weighing. With large enough scales 
and tanks the largest amounts of water used for engine tests 
may be readily handled. 

Measuring Water. — When it is not convenient to weigh 



288 THERMODYNAMICS OF THE STEAM-ENGINE. 

water directly it may be measured in tanks or other receptacles 
of known volume. Commonly two are used, so that one may 
fill while the other is emptied. The volume of a receptacle 
may be calculated from its dimensions or may be determined 
by weighing in water enough to fill it. When desired a 
receptacle may be provided with a scale showing the depth of 
the water, and so partial volumes can be determined. A closed 
receptacle may be used to measure hot water or other fluids. 
Water metres are of two kinds: some measure the 
water by the displacement of a piston or of pistons ; in others the 
water is recorded by a rotating piece which is commonly made 
of a material having nearly the density of water. The latter 
may serve to check the use of water drawn from a city water- 
supply, but they are seldom reliable enough for use in engine- 
testing. Piston water-metres can be made to give almost any 
desired degree of accuracy ; good commercial piston-metres 
seldom have more than one or two per cent of error. They 
should always be tested under the conditions of the test, 
taking account of both the amount and the temperature of 
the water measured. Tests of marine engines can hardly be 
made without the aid of a metre. For such tests the metre 
may be placed on a by-pass through which the feed-water 
from the surface- condenser can be made to pass by closing a 
valve on the direct line of feed-pipe. If necessary the metre 
can be quickly shut off and the direct line can be opened. 
The chief uncertainty in the use of a metre is due to air in 
the water; to avoid error from this source the metre should 
be frequently vented to allow air to escape without being 
recorded by the metre. 

Weirs and Orifices. — So far as possible the use of weirs 
and orifices for water during engine tests should be avoided. 
for, in addition to the uncertainties unavoidably connected 
with such hydraulic measurements, difficulties are liable to 
arise from the temperature of the water and from the oil in it. 
A very little oil is enough to sensibly affect the coefficient for 
a weir or orifice. The water flowing from the hot-well of a 



TESTING STEAM-ENGINES. 289 

jet-condensing engine is so large in amount that it is deemed 
advisable to measure it on a weir; the effect of temperature 
and oil is less than when the same method is used for 
measuring condensed steam from a surface-condenser. 

Calorimeters. — When superheated steam is supplied to 
an engine it is sufficient to take the temperature of the steam 
in the steam-pipe near the engine. When moist steam is 
used the condition of the steam must be determined by 
a calorimetric experiment. Four kinds of calorimeters will 
be described out of a large number that have been used 
by different experimenters and at different times. They are 
the barrel calorimeter, the Barrus continuous water-calorim- 
eter, the throttling-calorimeter, and the separating calorim- 
eter. 

The Barrel Calorimeter. — A wooden barrel set on scales 
is provided with a large valve for emptying it, and provision 
is made for filling it with cold water, usually from a hydrant- 
pipe, and for bringing the steam to be tested. Some form 
of stirrer must be used, a good form being a wooden propeller- 
wheel on a wooden shaft with a hand- crank. 

The method of making a test is as follows : The barrel is 
weighed empty, and a suitable quantity of cold water is run in 
and weighed. The temperature of the cold water should be 
taken as it enters. The steam-pipe usually terminates in a 
piece of rubber hose which may be swung into or out of the 
barrel. When the barrel is nearly filled with cold water the 
steam-valve may be opened until all condensed water is blown 
from the pipe and the hose is warmed up ; then the hose may 
be swung into the barrel and steam may be run into the water 
till a proper amount is condensed. A preliminary calculation 
will determine the proper weights of water and steam to give 
a good range of temperatures in the calorimeter. After the 
steam is run in the water in the barrel may be well stirred 
and the highest temperature taken as the final temperature. 

To eliminate the action of the wood of the barrel one or 
more tests are made and rejected, and the times of running in 



2g0 THERMODYNAMICS OF THE STEAM-ENGINE. 

water and steam are made equal, so that the barrel, which is 
already warmed by the preceding test, may give up as much 
heat during one part of the process as it receives during the 
other part. 

If the pressure of the steam is/, and the part of each pound 
of the mixture which is steam is represented by ;r, while the 
initial and final temperatures of the water are t 1 and t 9 , and 
the weights of the water and steam are W and w, then 

w(xr + q — q t ) = W(q % - q x ) ; 



x 



wr 



(265) 



r and q being the latent heat and heat of the liquid for the 
pressure /, and q x and q % being the heats of the liquid for the 
temperatures t x and t 3 . 

For example, suppose that 180 pounds of water at the tem- 
perature of 6o°.2 F. are run into a barrel calorimeter, and that 
the final temperature of the water in the calorimeter is I03°.6 
F., after J\ pounds of steam at 73.8 pounds by the gauge are 
run in and condensed. At an absolute pressure of 88.5 
pounds r= 890.4, q = 288.8; the heats of the liquid at 
6o°.2 and I03°.6 are 28.32 and 71.6. 

180(71.6 - 28.32) — 7.25(288.8 — 71.6) 
x — \L 3—L — * J v L 1 = 0.963 ; 

7.25 X 890.4 y ,5 ' 

consequently the per cent of priming is 3.7. 

It is to be remarked of this kind of calorimeter that satis- 
factory results are difficult to attain even when every care and 
precaution are used, and that a small error in determining the 
weight of steam, which is obtained by subtraction, makes a 
large difference in the result. 

Continuous Water-calorimeter. — The difficulty of ob- 
taining the weight of steam with sufficient accuracy which 
occurs in the use of the barrel calorimeter is avoided in the use 
of the continuous water-calorimeter, represented by Fig. 68. 
This calorimeter is essentially a small surface-condenser of 



TESTING STEAM-ENGINES. 



29I 



special form, so arranged that the condensed steam is weighed 
separately from the cooling water. 

Steam is brought to the calorimeter by the pipe j\ with the 
gauge i for giving the pressure. The pipe a, which forms the 



COLO WATER 




CONDENSED 
WATER 



Fig. 68. 

condensing surface, and which may conveniently be made of 
brass pipe one inch in diameter, should have the joints, above 
and below, clear of the bucket containing the cooling water. 
Steam is let into the pipe a at full boiler-pressure, and the con- 
densed water gathers in the pipe below, where the water-level 
is shown at e. The height of the water at e is kept constant 
by aid of the valve at d, which may have a long wooden handle 



292 THERMODYNAMICS OF THE STEAM-ENGINE. 

attached for convenient regulation. At h there is a thermome- 
ter to determine the temperature of the condensed steam. 
Since this temperature is only a little less than that due to the 
boiler-pressure, the condensed water should be led through a 
cooler like a simple surface-condenser, with a separate stream 
of cooling water, and the cooled water may be collected and 
weighed on suitable scales. 

The cooling water for the calorimeter is brought by the 
pipe b with a valve for regulating the supply, and is led away 
to a barrel on scales by the pipe c y with a valve to regulate the 
height of the water in the bucket. To insure a good circula- 
tion and a proper mingling of the cooling water the current is 
directed through a rubber hose to the bottom of the inner 
cylinder around the pipe a, thence up and into the top of 
the outer cylinder, thence down and out at the bottom of this 
cylinder and over a weir at the exit. The temperatures of the 
cooling water at entrance and exit are taken by the thermom- 
eters f and g, which should be reliable to -^ of one degree 
Fahrenheit. 

The pipe j leading to the calorimeter and the pipe con- 
taining the condensed steam should be well wrapped as far as 
to the valve at d. At k there is a brass cone to protect the 
covering of the pipe from water. 

Though not essential, it is convenient to line the bucket 
with sheet metal. 

In preparing for a test the water and steam are let on 
and properly regulated, and the calorimeter is allowed to 
run till all parts may be assumed to be at a constant tem- 
perature ; the cooling water from c and the condensed steam 
are then directed into the receptacles for weighing, and the 
time is noted as the beginning of the test. The steam-pres- 
sure and the several temperatures are taken at intervals and 
recorded. At the end of half an hour or an hour the 
cooling water and condensed water are diverted from the 
weighing receptacles, and the time is noted as the end of the 
test. The quantities of the cooling and condensed water can 



TESTING STEAM-ENGINES. 



293 



be weighed at the end of the test, or the test may be made 
continuous for any desired length of time by having two 
weighing receptacles for each, and filling and emptying them 
alternately. 

The radiation in thermal units per hour must be deter- 
mined by running the calorimeter without cooling water and 
with the bucket filled with hair-felt. 

In this or any form of calorimeter that is capable of giving 
accurate results it is essential that the steam-pressure should 
not change during a test, since a considerable change of pres- 
sure will vitiate the results on account of the heat absorbed 
or yielded by the pipes leading to the condenser. 

Let Wand w be the weights of the cooling water for the 
test, and let/ be the steam-pressure and / 3 the final tempera- 
ture of the condensed steam taken by the thermometer at h, 
while t 1 and / 2 are the initial and final temperatures of the cool- 
ing water; finally, let the radiation during the test be e ther- 
mal units. 



Then 



w (xr -\-q - g M )= IV(^ — q x ) + <?; 



x = 



wr 



(266) 



Example. — The following are the data of a test made in 
the laboratory of the Institute of Technology : 

Initial temperature of cooling water, . 37 . 49 F. 

Final 

Temperature of condensed steam, 

Pressure of the atmosphere, 

Pressure of steam by gauge, 



Duration of test, 
Radiation per hour, 
Weight of cooling water, . 
" " condensed water, 



8 3 u .8 4 F. 
304°.88 F. 
14.8 lbs. per sq. in. 

72.4 " " " " 
40 minutes. 

180 B. T. U. 
573.5 pounds. 
29.89 " 



x 



= 573»5(5i-9i ~ 5-53)+ 120-29.89(287.6- 274.4) . 

29.89 X 891.2 
x = 0.988. 
Per cent of priming, 1.2. 



294 



THERMODYNAMICS OF THE STEAM-ENGINE. 



It is apparent that any surface-condenser may be used in 
the same manner as a calorimeter, except that it is not 
usually convenient to fill such a condenser with steam at 
boiler-pressure. Since the wire-drawing of steam in a well- 
wrapped valve is accompanied with little loss of heat, this 
need not interfere with such a use of a condenser. In 
an engine test the quality of exhaust-steam flowing to a 
jet or a surface condenser can be determined by equation 
(265) or (266), except that the external radiation cannot 
always be satisfactorily determined. 

Throttling-calorimeter — A simple form of calorimeter, 
devised by the author, is shown by Fig. 69, where A is a 

reservoir about four inches in diameter 
and about 10 inches long to which 
steam is admitted through a half-inch 
pipe by with a throttle valve. near the 
reservoir. Steam flows away through 
an inch pipe d. At f is a gauge for 
measuring the pressure, and at e there 
is a deep cup for a thermometer to 
measure the temperature. The boiler- 
pressure may be taken from a gauge 
on the main steam-pipe near the 
calorimeter. It should not be taken 
from a pipe in which there is a rapid 
flow of steam as in the pipe b, since 
the velocity of the steam will affect the 
gauge-reading, making it less than the 
real pressure. The reservoir is wrapped 
with hair-felt and lagged with wood to reduce radiation of 
heat. 

When a test is to be made, the valve on the pipe d is 
opened wide (this valve is frequently omitted) and the valve 
at b is opened wide enough to give a pressure of five to fifteen 
pounds in the reservoir. Readings are then taken of the 
boiler-gauge, of the gauge at f y and of the thermometer at e. 




Fig. 69. 






TESTING STEAM-ENGINES. 295 

It is well to wait about ten minutes after the instrument is 
started before taking readings so that it may be well heated. 
Let the boiler-pressure be /, and let r and q be the latent 
heat and heat of the liquid corresponding. Let p 1 be the 
pressure in the calorimeter, and \ and t x the total heat and 
the temperature of saturated steam at that pressure, while t s 
is the temperature of the superheated steam in the calorim- 
eter. Then 

xr-\-q — \ + c p (t t — *,) ; 

.-. X= X > + C >& -'.)-? . . . . (2 6 7 ) 

r 

Example. — The following are the data of a test made 
with this calorimeter: 

Pressure of the atmosphere 14.8 pounds; 

Steam-pressure by gauge 69.8 " 

Pressure in the calorimeter, gauge... 12.0 (i 
Temperature in the calorimeter 268°.2 F. 

1 15 6.4 + 0.48(268.2 - 243,9) - 28 5-3 88 . 

892.7 " ' 9 ' 

Per cent of priming, 1.2. 

A little consideration shows that this type of calorimeter 
can be used only when the priming is not excessive ; otherwise 
the throttling will fail to superheat the steam, and in such 
case nothing can be told about the condition of the steam 
either before or after throttling. To find this limit for any 
pressure t s may be made equal to t l in equation (267) ; that is, 
we may assume that the steam is just dry and saturated at that 
limit in the calorimeter. Ordinarily the lowest convenient 
pressure in the calorimeter is the pressure of the atmosphere, or 
14.7 pounds to the square inch. The table following has been 
calculated for several pressures in the manner indicated. It 
shows that the limit is higher for higher pressures, but that 
the calorimeter can be applied only where the priming is 
moderate. 

When this calorimeter is used to test steam supplied to a 
condensing-engine the limit may be extended by connecting 



296 



THERMODYNAMICS OF THE STEAM-ENGINE. 



the exhaust to the condenser. For example, the limit at 100 
pounds absolute, with 3 pounds absolute in the calorimeter, is 
0.064, instead of 0.046 with atmospheric pressure in the calo- 
rimeter. 

LIMITS OF THE THROTTLING CALORIMETER. 



1 

Pressure. 








Priming. 






Absolute. 


Gauge. 




300 


285.3 


O.077 


250 


235-3 


O.070 


200 


185.3 


0.061 


175 


160.3 


O.058 


I50 


135.3 


O.052 


125 


no. 3 


O.046 


IOO 


85-3 


O.040 


75 


60.3 


O.032 


50 


35-3 


O.023 



In case the calorimeter is used near its limit — that is, when 
the superheating is a few degrees only — it is essential that the 
thermometer should be entirely reliable; otherwise it might 
happen that the thermometer should show superheating when 
the steam in the calorimeter was saturated or moist. In any 
other case a considerable error in the temperature will produce 
an inconsiderable effect on the result. Thus at 100 pounds 
absolute with atmospheric pressure in the calorimeter io° F. 
of superheating indicates 0.035 priming, and 15 F. indicates 
0.032 priming. So also a slight error in the gauge-reading has 
little effect. Suppose the reading to be apparently 100.5 
pounds absolute instead of 100, then with io° of superheat- 
ing the priming appears to be 0.033 instead of 0.032. 

It has been found by experiment that no allowance need 
be made for radiation from this calorimeter if made as described, 
provided that 200 pounds of steam are run through it per 
hour. Now this quantity will flow through an orifice one- 
fourth of an inch in diameter under the pressure of 70 pounds 
by the gauge, so that if the throttle-valve be replaced by such 
an orifice the question of radiation need not be considered. 
In ^uch case a stop-valve will be placed on the pipe to shut 



TES TING S TEA M-ENGINES. 



29/ 



off the calorimeter when not in use ; it is opened wide when a 
test is made. If an orifice is not provided the throttle-valve 
may be opened at first a small amount, and the temperature 
in the calorimeter noted ; after a few minutes the valve may be 
opened a trifle more, whereupon the temperature may rise, if 
too little steam was used at first. If the valve is opened little 
by little till the temperature stops rising it will then be certain 
that enough steam is used to reduce the error from radiation 
to a very small amount. 

Various modifications of the throttling-calorimeter have 
been proposed, mainly with a view to reducing its size and 
weight. Almost any of them will prove satisfactory in prac- 
tice, but some will be found to be liable to error from radia- 
tion or from the fact that there is not sufficient opportunity 
for the steam to come to rest and properly develop the super- 
heating due to throttling. 

Separating Calorimeter. — If steam contains more than 
three per cent of moisture the priming 
may be determined by a good sepa- 
rator which will remove nearly all the 
moisture. It remains to measure the 
steam and water separately. The water 
may be best measured in a calibrated 
vessel or receiver, while the steam may 
be condensed and weighed, or may be 
gauged by allowing it to flow through 
an orifice of known size. A form of 
separating calorimeter devised by Prof. 
Carpenter* is shown by Fig. 70. 

Steam enters a space at the top 
which has sides of wire gauze and a 
convex cup at the bottom. The water 
is thrown against the cup and finds its 
way through the gauze into an inside 
chamber or receiver and rises in a FlG - 70. 




* Trans. Am. Soc. Mech. Engs., vol. xvii, p. 608. 



298 THERMODYNAMICS OF THE STEAM-ENGINE. 

water-glass outside. The receiver is calibrated by trial, so 
that the amount of water may be read directly from a gradu- 
ated scale. The steam meanwhile passes into the outer cham- 
ber which surrounds the inner receiver and escapes from an 
orifice at the bottom. The steam may be determined by con- 
densing, collecting, and weighing it ; or it may be calculated 
from the pressure and the size of the orifice. When the steam 
is weighed there is no radiation error, since the inner cham- 
ber is protected by the steam in the outer chamber. This 
instrument may be guarded against radiation by wrapping 
and lagging, and then if steam enough is used the radiation 
will be insignificant, just as was found to be the case for the 
throttling-calorimeter. 

There is a question whether this instrument or the throt- 
tling-calorimeter should be called a calorimeter; perhaps they 
may be better considered to be priming-gauges. 

Method of Sampling Steam — It is customary to take a 
sample of steam for a calorimeter or priming-gauge through a 
small pipe leading from the main steam-pipe. The best 
method of securing a sample is an open question ; indeed, it 
is a question whether we ever get a fair sample. There is no 
question but that the composition of the sample is correctly 
shown by any of the calorimeters described, when the 
observer makes tests with proper care and skill. It is 
probable that the best way is to take steam through a 
pipe which reaches at least halfway across the main steam- 
pipe, and which is closed at the end and drilled full of 
small holes. It is better to have the sampling-pipe at the side 
or top of the main, or otherwise arranged so that water tric- 
kling along the bottom of the main shall not enter the calo- 
rimeter. Again, it is better to take a sample from a pipe 
through which steam flows vertically upward. The sampling- 
pipe should be short and well wrapped to avoid radiation. 

Indirect Engine Test — It is often difficult, if not impos- 
sible, to determine the steam used by an engine directly if it 
has not a surface-condenser, and if at the same time the boiler 



TESTING STEAM-ENGINES. 299 

or boilers supplying it with steam do other work also. If the 
engine has a jet-condenser then the overflow from the hot-well 
may be determined by allowing it to flow over a weir. At the 
same time the engine may be indicated and the temperatures 
of the cold injection-water and of the overflow from the hot- 
well may be determined. If possible the priming in the 
steam-pipe should be determined by aid of a throttling calo- 
rimeter, and if the engine has a steam-jacket the radiation may 
be found from the condensation of steam in the jackets while 
the engine is at rest. For approximate work the steam in the 
supply-pipe may be assumed to be dry, and the radiation may 
be inferred from comparison with an engine of the same size 
and type; or, since the radiation is nearly constant, it may be 
treated as a constant unknown error. 

Suppose that the engine usesiJ/ pounds of steam per horse- 
power per hour, and that G pounds of condensing water are 
used per pound of exhaust-steam ; then 

M(i + G) = A (268) 

is equal to the overflow of the hot-well in pounds per hour as 
determined by the aid of a weir. 

If the injection-water comes in at the temperature t { and 
flows from the hot-well with the condensed steam at the 
temperature t k , then the heat taken up by the condensing 
water is 

MG(q k - &), 

where q k and q t are the heats of the liquid corresponding to the 
temperatures t k and t i% The heat radiated per hour may be 
represented by R. The heat changed into work will be 

60 H.P. 



33000 



where H.P. represents the horse-power of the engine deter- 
mined from the indicator-diagrams. 



300 THERMODYNAMICS OF THE STEAM-ENGINE. 

The heat supplied to the engine per hour will be 

M(xr + q-q k ), 

where r and q are the heat of vaporization and heat of the 
liquid at the pressure/ in the steam-pipe, and x is the quality 
of the steam'; usually, x is between 0.98 and unity. This 
equation assumes that the feed-water is drawn from the hot- 
well at the temperature t k . 

But the heat supplied to the engine is equal to the heat 
accounted for as work, by radiation, and as carried away by 
the cooling water, so that we have 

M{xr + g - ft) = 6 ° ^ + R + MG(<? t - ft). (269) 

From equations (268) and (269) we have 

60 H.P. _ ., 
h R + A{g k - q t ) 

m= 33 °°° ^ :.. . . ( 27 o) 

xr + q — qi 

by which the steam per horse-power can be readily calculated, 
and the thermal units per horse-power per minute can be found 
from the expression 

M(xr + q-q k )+6o (271) 



CHAPTER XIV. 

INFLUENCE OF THE CYLINDER WALLS. 

The difference between the action of steam in the cylinder 
of a steam-engine and the calculated action of steam in a non- 
conducting cylinder is due mainly to the influence of the 
walls of the cylinder, on which steam condenses during admis- 
sion and from which water is vaporized during expansion and 
exhaust. This influence is so intense, and at the same time 
so complicated, that any attempt to base the design of a steam- 
engine on the theoretical discussion of the cycle for a non-con- 
ducting engine leads only to confusion and disappointment. 
This became evident as soon as an attempt was made to use 
the results of the thermodynamic investigations of Rankine 
and Clausius for that purpose, and in consequence the whole 
thermodynamic treatment of the properties of steam and of its 
application to the production of power was looked upon with 
disfavor by engine-builders. Tests on engines showed an 
actual steam-consumption of quarter or half more than was 
given by theoretical calculations; and, again, the conditions 
which the theory indicated as favorable to economy often 
gave the worst results. This discrepancy between theory and 
practice was most notable in the case of cut-off and expansion 
in the cylinder. Thus on page 238 it is shown that 
complete expansion to the back-pressure gives a higher effi- 
ciency than incomplete expansion with a drop at release. But 
many experiments, for example those by Isherwood on the 
U. S. S. Michigan in 1 861, showed conclusively that the best 
economy was obtained by use of a moderately short cut-off. 

301 



302 



THERMODYNAMICS OF THE STEAM-ENGINE. 



U.S. S. MICHIGAN 
Abscissae per cents of cut off 
Ordinates pounds of steam 
per horse power per hour. 



40 




From the accompanying table of conditions and results it 
appears that for the engines of the Michigan the best econ- 
omy in steam-consumption is given for a cut-off at \ of 
the stroke. The same result is shown even more clearly by a 
diagram in which each cut-off is laid off as an abscissa and the 
corresponding steam-consumption is laid off as an ordinate, as 

shown by Fig. Ji. 
To make the dia- 
gram clear and com- 
pact, the axis of ab- 
scissae is taken at 30 
pounds of steam per 
horse- power per 
hour. An inspection 
of this diagram and 
of the figures in the 
table show a regu- 
^ larity in the results 
which can be at- 
tained only when 
tests are made with care and skill. The only condition pur- 
posely varied is the cut-off; the only condition showing im- 
portant accidental variation is the vacuum, and consequently 
the back-pressure in the cylinder. To allow for the small 
variations in the back-pressure Isherwood changed the mean 
effective pressure for each test by adding or subtracting, as 
the case might require, the difference between the actual 
back-presure and the mean back-pressure of 2.7 pounds per 
square inch, as deduced from all the tests. 

An inspection of any such a series of tests having a wide 
range of expansions will show that the steam-consumption 
decreases as the cut-off is shortened till a minimum is reached, 
usually at ^ to \ stroke ; any further shortening of the cut-off 
will be accompanied by an increased steam-consumption, 
which may become excessive if the cut-off is made very short. 
Some insight into the reason for this may be had from the 



30 



0.2 



0.1 

Fig. 



0.6 



0.8 



7i. 



INFLUENCE OF THE CYLINDER WALLS. 



303 



Table II. 

TESTS ON THE ENGINE OF THE U. S. S. MICHIGAN. 

CYLINDER DIAMETER, 36 INCHES ; STROKE, 8 FEET. 

By Chief-Engineer Isherwood, Researches in Experimental Steam 

Engineering. 



Duration, hours 

Cut-off 

Revolutions per minute 

Boiler-pressure, pounds per sq. in. above 

atmosphere 

Barometer, inches of mercury 

Vacuum, inches of mercury 

Steam per horse-power perhour, pounds 
Percent of water in cylinder at release. 



I. 


II. 
72 


III. 
72 


IV. 

72 


V. 

72 


VI. 

72 


72 


11/12 


7/10 


4/9 


3/10 


i/4 


1/6 


20.6 


15.6 


17-3 


137 


13-9 


II. 2, 


21.0 


19 5 


21.0 


21.0 


21.0 


21 .O 


30.IJ29-8 


29.7 


30.1 


29-9 


29-9 


26.5I26.1 


26.3 


25.8 


25.8 


25.6 


38.0I33.8 


32.7 


34.7-34.5136.8 


10.7 


15-3 


27.2 


41.739.6 

1 


42. 1 



VII. 

72 

4/45 



22.0 

29.9 

24.1 

41.4 

45- 1 



per cent of water in the cylinder, calculated from the dimen- 
sions of the cylinder and the pressures in the cylinder taken 
from the indicator-diagram. The method of the calculation 
will be given in detail a little later in connection with Hirn's 
analysis. It will be sufficient now to notice that the amount 
of water in the cylinder of the engine of the Michigan at 
release increased from 10.7 per cent for a cut-off at \^ of the 
stroke to 45.1 per cent for a cut-off at T 4 r of the stroke. 
Now all the water in the cylinder at release is vapor- 
ized during the exhaust, the heat for this purpose being 
abstracted from the cylinder walls, and the heat thus 
abstracted is wasted, without any compensation. The walls 
may be warmed to some extent in consequence of the rise of 
pressure and temperature during compression, but by far the 
greater part of the heat abstracted during exhaust must be 
supplied by the incoming steam at admission. There is 
therefore a large condensation of steam during admission and 
up to cut-off, and the greater part of the steam thus con- 
densed remains in the form of water and does little if any- 
thing toward producing work. This may be seen by inspec- 
tion of the table of results of Dixwell's tests on page 371. 
With saturated steam and with cut-off at 0.217 of the 
stroke, 52.2 per cent of the working substance in the cylinder 



304 THERMODYNAMICS OF THE STEAM-ENGINE. 

was water. Of this 19.8 per cent was reevaporated during 
expansion and 32.4 per cent remained at release to be re- 
evaporated during exhaust. When the cut-off was lengthened 
to 0.689 of the stroke, there was 27.9 per cent of water at 
cut-off and 23.9 per cent at release. The statement in per- 
centages gives a correct idea of the preponderating influence 
of the cylinder walls when the cut-off is unduly shortened ; it 
is, however, not true that there is more condensation with a 
short than with a long cut-off. On the contrary there is more 
water condensed in the cylinder when the cut-off is long, only 
the condensation does not increase as fast as do the weight of 
steam supplied to the cylinder and the work done, and conse- 
quently the condensation has a less effect. 

Hirn's Analysis — Though the method just illustrated 
gives a correct idea of the influence of the walls of the 
cylinder of a steam-engine, our first clear insight into the 
action of the walls is due to Hirn,* who accompanied his 
exposition by quantitative results from certain engine tests. 
The statement of his method which will be given here is de- 
rived from a memoir by Dwelshauvers-Dery.f 

Let Fig. 72 represent the cylinder of a steam-engine and 
the diagram of the actual cycle. For sake of simplicity the 

diagram is . represented without lead of 
admission or release, but the equations to 
be deduced apply to engines having either 
or both. The points 1, 2, 3, and o are 
the points of cut-off, release, compression, 
and admission. The part of the cycle 
from o to 1, that is, from admission to 




I 



JT 



F ~~" cut-off, is represented by a ; in like man- 

ner, b, c, and d represent the parts of 
the cycle during expansion, exhaust, and compression. The 
numbers will be used as subscripts to designate the properties 

* Bulletin de la Soc. Ind. de Mulhouse, 1873 ; The'orie Me'chanique de la 
Chaleur, vol. ii., 1876. 

f Revue universelle des Mines, vol. viii. p. 362, 1880. 



INFLUENCE OF THE CYLINDER WALLS. 30$ 

of the working fluid under the conditions represented by the 
points indicated, and the letters will be used in connection 
with the operations taking place during the several parts of 
the cycle. Thus at cut-off the pressure is p lt and the tem- 
perature, heat of the liquid, heat of vaporization, condition, 
etc., are represented by t lt q lf r 1} x v etc. The external work 
from cut-off to release is W b , and the heat yielded by the 
walls of the cylinder due to reevaporation is Q b . 

Suppose that M pounds of steam are admitted to the 
cylinder per stroke, having in the supply-pipe the pressure p 
and the condition x ; that is, each pound is x part steam 
mingled with I — x of water. The heat brought into the 
cylinder per stroke, reckoned from freezing-point, is 

Q=M(q + xr) (272) 

Should the steam be superheated in the supply-pipe to 
the temperature t si then 

Q = M[\ + c t {t s -t)l .... (273) 

in which c p = 0.4805 is the specific heat of superheated steam 
at constant pressure. 

Let the heat-equivalent of the intrinsic energy of the entire 
weight of water and steam in the cylinder at any point of the 
cycle be represented by I ; then at admission, cut-off, release, 
and compression we have 

I = MXq* + *opo) ; (274) 

/^(M+MJfa+x^); . . . (275) 

/„ = (M + M t )(q t + x iPi ) ; . . . . (276) 

I % = M (q z + ^p 3 ); {277) 

in which p is the heat-equivalent of the internal work due to 
vaporization of one pound of steam, and M is the weight of 
water and steam caught in the cylinder at compression, cal- 
culated in a manner to be described hereafter. 

If the steam is superheated at any point of the cycle 
the corresponding intrinsic energy may be calculated by aid of 



306 THERMODYNAMICS OF THE STEAM-ENGINE. 

equation (179), page 133. For example the heat-equivalent of 
the intrinsic energy at release may be 



/, = {M + M a ) J ^— p,(v, - s,) + q, + p A , ( 



278) 



where v^ is the specific volume of the superheated steam and 
s t is the specific volume of saturated steam at the pressure p^. 

k is, of course, the constant — , as given on page 129. 

At admission the heat-equivalent of the fluid in the 
cylinder is 7 , and the heat supplied by the entering steam up 
to the point of cut-off is Q. Of the sum of these quantities a 
part, A W a , is used in doing external work, and a part remains 
as intrinsic energy at cut-off. The remainder must have been 
absorbed by the walls of the cylinder, and will be represented 
by Q a . Hence 

0. = Q + /.-/, -A W a . 

From cut-off to release the external work Wb is done, and 
at release the heat-equivalent of the intrinsic energy »*s /,. 
Usually the walls of the cylinder, during expansion, supply 
heat to the steam and water in the cylinder. To be more 
explicit, some of the water condensed on the cylinder walls 
during admission and up to cut-off is evaporated during expan- 
sion. This action is so energetic that 7 2 is commonly larger 
than I x . Since heat absorbed by the walls is given a positive 
sign, the contrary sign should be given to heat yielded by 
them ; it is, however, convenient to give a positive sign to 
all the interchanges of heat in the equations, and then in 
numerical problems a negative sign will indicate that heat is 
yielded during the operation under consideration. For expan- 
sion, then, 

Q b = I, - /, - A W b . 

During the exhaust the external work W c is done by the 
engine on the steam, the water resulting from the condensation 



INFLUENCE OF THE CYLINDER WALLS. 307 

of the steam in the condenser carries away the heat Mq Ai the 
cooling water carries away the heat G{q k — q t ), and there 
remains at compression the heat-equivalent of intrinsic energy 
7 3 . So that 

Qc = /, - 7 3 - Mq, - G(q k - q t ) + A W e , 

in which q A is the heat of the liquid of the condensed steam, 
and G is the weight of cooling water per stroke which has on 
entering the heat of the liquid q £ , and on leaving the heat of 
the liquid q k . 

During compression the external work W d is done by the 
engine on the fluid in the cylinder, and at the end of com- 
pression, i.e., at admission, the heat-equivalent of the intrinsic 
energy is 7 o . Hence 

. a, = /,-/,+ aw* 

It should be noted (Fig 72) that the work W a is repre- 
sented by the area which is bounded by the steam line, the 
ordinates through o and 1 and by the axis OV. And in like 
manner the works JV&, W c , and W d are represented by areas 
which extend to the axis OV. In working up the analysis from 
a test the line of absolute zero of pressure may be drawn 
und^r the atmospheric line as in Fig. 
73, or proper allowance may be made 
after the calculation has been made 
with reference to the atmospheric 

* 111C * J ' Atmospheric line 

For convenience these four equa- y . 

tions will be assembled as follows : Fig. 73. 

Q.= Q+I.-I t -AW.; (279) 

Q h = /,-/,- AW„; (280) 

Q c = I,-I,-M„-G{q k -q,) + AW c ; . (281) 

Qt=T,-I. + AW d (282) 




308 THERMODYNAMICS OF THE STEAM-ENGINE. 

A consideration of these equations shows that all the 
quantities of the right-hand members can be obtained directly 
from the proper observations of an engine test except the 
several values of /, the heat-equivalents of the intrinsic ener- 
gies in the cylinder. These quantities are represented by 
equations (274) to (277), in which there are five unknown 
quantities, namely x , x lf x„ x 3f and M o . Should the steam 
be superheated at any point, as at release, the proper modifi- 
cation must be made as indicated by equation (278), in 
which case we have the unknown v^ instead of x„ 

Let the volume of the clearance-space between the valve 
and the piston when it is at the end of its stroke be V ; and 
let the volumes developed by the piston up to cut-off and 
release be V l and F" 2 ; finally, let V 3 represent the correspond- 
ing volume at compression. The specific volume of one pound 
of mixed water and steam is 

v = xu -f- 0", 

and the volume of M pounds is 

V= Mv = M(xu + cr). 

At the points of admission, cut-off, release, and compres- 
sion 

V = M (x u + a) ; (283) 

V + V x = (M + MX**u x + &) ; . . (284) 

V +K=(M+M )(x,u, + <t); . . (285) 

V.+ K = M 9 (x,u,+ <r) (286) 

There is sufficient evidence that the steam in the cylinder 
at compression is nearly if not quite dry, and as there is com- 
paratively little steam present at that time, there cannot be 
much error in assuming 

x. = I . 



INFLUENCE OF THE CYLINDER WALLS. 309 

This assumption gives, by equation (286^, 
V 4- V V 4- V 

in which y % is the density or weight of one cubic foot of dry 
steam at compression. 

Applying this result to equations (283) to (286) gives 

*. = -—*- -; (288) 



x, = 



r. + K *. 



(289) 



We are now in condition to find the values of I , I lf / 2 , and 
7 3 , and consequently can calculate all the interchanges of heat 
by equations (279) to (282). 

If the steam is superheated at some point, as at release, 
proper allowance must of course be made. Thus we may 
have 

V a +V, = {M+M t )V, 

or 

v - V >+ K ( \ 

*~M+M ' ' ' ' * {29I) 

which gives the means of calculating 7 2 by equation (278). 
The fact that steam is superheated will be indicated by an 
apparent value of x greater than unity when calculated by one 
of the equations (288), (289), or (290). It is probable that 
any test of an engine with much compression will give super- 
heated steam at admission when x^ is calculated by equation 
(288). Recent tests by Callandar and Nicolson confirm the 
conclusions from such calculations. There will, however, be 
little error in most cases from assuming x to be unity as well 



3IO THERMOD YNAMICS OF THE STEAM-ENGINE. 

as x z , and the work will be a little simplified by such an assump- 
tion. It may be remarked in passing that when the specific 
volume and pressure of superheated steam are known the 
temperature may be calculated by equation (177), page 130. 
There is, however, no reason except general interest for 
making such a calculation. 

In the diagram, Fig. 72, the external work during exhaust 
is all work done by the piston on the fluid, since the release is 
assumed to be at the end of the stroke. If the release occurs 
before the end of the stroke, some of the work, namely, from 
release to the end of the stroke, will be done by the steam 
on the piston, and the remainder, from the end of the stroke 
back to compression, will be done by the piston on the fluid. 
In such case W e will be the difference between the second and 
the first quantities. If an engine has lead of admission, a 
similar method may be employed ; but at that part of the 
diagram the curves of compression and admission can be dis- 
tinguished with difficulty, if at all, and little error can arise 
from neglecting the lead. 

The several pressures at admission, cut-off, release, and 
compression are determined by the aid of the indicator-dia- 
gram, and the pressures in the steam-pipe and exhaust-pipe or 
condenser are determined by gauges. The weight M of steam 
supplied to the cylinder per stroke is best determined by con- 
densing the exhaust-steam in a surface-condenser and collect- 
ing and weighing it in a tank. If the engine is non-condens- 
ing, or if it has a jet-condenser, or if for any reason this 
method cannot be used, then the feed-water delivered to the 
boiler may be determined instead. The cooling or condensing 
water, either on the way to the condenser or when flowing 
from it, may be weighed, or for engines of large size may be 
measured by a metre or gauged by causing it to flow over a 
weir or through an orifice. The several temperatures t t iy 
and t k must be taken by proper thermometers. When a jet- 
condenser is used, and the condensing water mingles with the 
steam, / 4 is identical with t k . The quality x of the steam in 



INFLUENCE OF THE CYLINDER WALLS. 311 

the supply-pipe must be determined by a steam-calorimeter. 
A boiler with sufficient steam-space will usually deliver nearly 
dry steam ; that is, x will be nearly unity. If the steam is 
superheated, its temperature t s may be taken by a ther- 
mometer. 

Let the heat lost by radiation, conduction, etc., be Q e ; 
this is commonly called the radiation. Let the heat supplied 
by the jacket be Qj. Of the heat supplied to the cylinder per 
stroke, a portion is changed into work, a part is carried away 
by the condensed steam and the cooling or condensing water, 
and the remainder is lost by radiation ; therefore 

Q,= Q+ Q-Mg-G(q k -g^-A( W a + W„- W c - W d ). (292) 

The heat Qj supplied by a steam-jacket may be calculated 
by the equation 

Q J = m(x'r' + q'-q"),. . . . (293) 

in which m is the weight of water collected per stroke from the 
jacket ; x' ', r' ', and q' are the quality, the heat of vaporization, 
and the heat of the liquid of the steam supplied; and q" is the 
heat of the liquid when the water is withdrawn. When the 
jacket is supplied from the main steam-pipe, x' is the same as 
the quality in that pipe. When supplied direct from the boiler, 
x' may be assumed to be unity. If the jacket is supplied 
through a reducing-valve, the pressure and quality may be 
determined either before or after passing the valve, since 
throttling does not change the amount of heat in the steam. 
Should the steam applied to the jacket be superheated from 
any cause, we may use the equation 

Q j = m\X-\-c p (t:-t')-q"-\, . . . (294) 

in which V is the total heat of saturated steam at the temper. • 
ture t' ', and tj is the temperature of the superheated steam. 

Equation (292) furnishes a method of calculating the heat 
lost by radiation and conduction ; but since Q e is obtained by 



312 THERMODYNAMICS OF THE STEAM-ENGINE. 

subtraction and is small compared with the quantities on the 
right-hand side of the equation, the error of this determination 
may be large compared with Q e itself. The usual way of de- 
termining Q e for an engine with a jacket is to collect the water 
condensed in the jacket for a known time, an hour for exam- 
ple, when the engine is at rest, and then the radiation of heat 
per hour may be calculated. If it be assumed that the rate 
of radiation at rest is the same as when the engine is running, 
the radiation for any test may be inferred from the time of the 
test and the determined rate. But the engine always loses 
heat more rapidly when running than when at rest, so that this 
method of determining radiation always gives a result which 
is too small. 

If a steam-engine has no jacket it is difficult or impossible 
to determine the rate of radiation. The only available way 
appears to infer the rate from that of some similar engine with 
a jacket. Probably the best way is to get an average value 
of Q e from the application of equation (292) to a series of care- 
fully made tests. 

It is well to apply equation (292) to any test before begin- 
ning the calculation for Hirn's analysis, as any serious error is 
likely to be revealed, and so time may be saved. 

When the radiation Q e is known from a direct determina- 
tion of the rate of radiation, we may apply Hirn's analysis to 
a test on an engine even though the quantities depending on 
the condenser have not been obtained. For from equation 
(292) 

-Mq K - G{q k - q t ) =Q e -Q- Q J + A(W a + W b ~W c - W d \ 

and consequently 

Q, = 1,-1, -Q-Qj+ Q. + A(W.+ m-2W e - W d ). (294) 

Thus it is possible to apply the analysis to a non-condensing 

engine or to the high-pressure cylinder of a compound engine. 

It is apparent that the heat Q c , thrown out from the walls 



INFLUENCE OF THE CYLINDER WALLS. 313 

of the cylinder during exhaust, passes without compensation 
to the condenser, and is a direct loss. Frequently it is the 
largest source of loss, and for this reason Hirn proposed to 
make it a test of the performance and perfection of the 
engine; but such a use of this quantity is not justifiable, and 
is likely to lead to confusion. 

The heat Q b that is restored during expansion is supplied 
at a varying and lower temperature than that of the source of 
heat, namely, the boiler, and though not absolutely wasted, 
is used at a disadvantage. It has been suggested that an 
early compression, as found in engines with high rotative 
speed, warms up the cylinder and so checks initial condensa- 
tion, thereby reducing Q a and finally Q e also. Such a storing 
of heat during compression and restoring during expansion is 
considered to act like the regenerator of a hot-air engine, and 
to make the efficiency of the actual cycle approach the 
efficiency of the ideal cycle more nearly than would be the 
case without compression. It does not, however, appear that 
engines of that type have exceeded, if they have equalled, 
the performance of slow-speed engines with small clearance 
and little compression. 

Application. — In order to show the method of applying 
Hirn's analysis the complete calculation for a test made on a 
srr 11 Corliss engine in the laboratory of the Massachusetts 
Institute of Technology will be given: 

Diameter of the cylinder 8 inches. 

Stroke of the piston 2 feet. 

Piston displacement : crank end. , . . 0.6791 cu. ft. 

head end. . . . . 0.7016 " 
Clearance, per cent of piston displacement : 

crank end 3.75 

head end 5.42 

Boiler-pressure by gauge 77.4 pounds. 

Barometer 14.8 " 

Condition of steam, two per cent of moisture. 



3H 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Events of the stroke : 

Cut-off : crank end 0.306 of stroke. 

head end 0.320 " 

Release at end of stroke. 

Compression : crank end.. 0.013 of stroke. 

head end 0.0391 " 

Duration of the test, one hour. 

Total number of revolutions 3692 

Weight of steam used 548 pounds. 

Weight of condensing water used.. . 14,568 " 
Temperatures : 

Condensed steam t i = 141 °.i F. 

Condensing water : cold t t = . 52°.9 F. 

warm t k = 88°.3 F. 



ABSOLUTE PRESSURES, FROM INDICATOR-DIAGRAMS, AND 
CORRESPONDING PROPERTIES OF SATURATED STEAM. 





Crank End. 


Head End. 




/ 


7 


P 


u 


P 


q 


p 


u 


Cut-off 

Release 

Compression. . 
Admission... . 


83.6 
29.2 
I4.8 
21.8 


284.6 
217.8 
l8l. I 
201.5 


813.O 

864.8 
893.2 

877-4 


5.190 
13.924 

26.464 

18.344 


83-3 
31-9 
14.8 
29.8 


284.4 
222.9 
181. 1 
219.0 


813.2 
860.8 
893.2 
863.9 


5.207 

12.804 
26 . 464 
13-664 



MEAN PRESSURES, AND HEAT- EQUIVALENTS OF EXTERNAL 

WORKS. 





Crank End. 


Head End. 




Mean Pressures. 


Equivalents of 
Work. 


Mean Pressures. 


Equivalents of 
Work. 




87.7 

44-5 
14.8 

18.3 


3-369 

3-877 
I.836 
O.0299 


89-3 
47.I 
I4.8 
21.8 


3-7II 
4-159 
1.847 
0.1 104 




Exhaust 


Compression 





VOLUMES, CUBIC FEET. 





Crank End. 


Head End. 


At cut-off, V + Ti 


O.2333 
O.7046 
O.0343 
O.02550 


O.2626 


At release, Va -j- Vi 


O.7396 
O.0655 


At compression, Vo -j- V% 


At admission, Vo 


O.03806 







INFLUENCE OF THE CYLINDER WALLS. 315 

At the boiler-pressure, 92.1 pounds absolute, we have 
r = 888.4, q = 291.7. 

The steam used per stroke is 

548 
M = 7 — = 0.0742 pound. 

2 x 3692 

The steam caught in the clearance space at compression, 
on the assumption that the steam is then dry and saturated, 
is obtained by multiplying the mean volume at that point by 
the weight of one cubic foot of steam at the pressure at com- 
pression, which is 0.03781 of a pound. 

0.0343 4- 0.06155 
.-. M = — ^- J — X 0.03781 =0.0019 of a pound; 

M-{- M = 0.0742 -f- 0.0019 = 0.0761 pound. 

The condensing water used per stroke is 

_ 14568 

G = —£— = 1.973. 

2 X 3692 ^ /J 

Q = M{xr -f- q) = 0.0742(0.98 X 888.3 + 291.8) = 86.243 ; 
-|(0.02 550 + 0.03 806) I 



0.0019 X ^18.344+ 13.664) 62.4Xi( x 8.344+i3- 66 4) 
= 1.043. 

This indicates that the steam is superheated at admission. 
Such may be the case, or the appearance may be due to an 
error in the assumption of dry steam at compression, or to 
errors of observation. It is convenient to assume x = I. 

_ ^0+ v, <l 



3l6 THERMODYNAMICS OF THE STEAM-ENGINE. 

_ i(o-2333 + 0-2626) I 

~ 0.0761 x 4(5-i90+ 5-207) "" 62.4 x K5-I90 + 5-207) 
= 0.6236. 

K+V, * 

x n = 



x„ 



4(0.7046-1-0.7396) 



0.0761 x 4(13-924+ 12.804) 62.4x4(13-924+12.804) 

= 0.7088. 

I = M (q + x p ) ; 

.'. /, = |X o.ooi9[20i.5 + 219.0 + 1.00(877.4 + 863.9)] 
= 2.054. 

I, = (M + MJfa + x lPl ) ; 

.-. 7, = ix 0.0761 [284.6 + 284.4 + 0.6236(813.0 + 813.2)] 
= 60.238. 

/^(M+MJfa + x^); 

.-. 7 2 = 4 X 0.076i[2i7.8 + 222.0 + 0.7088(864.8 + 861.8)] 
= 63.311. 

h = M£q t + * 3 p 3 ) ; 
.-. 7 S = 0.0019(181.1 + 893.2) = 2.041. 

Qa=Q + Io-f 1 -AW a ; 
.-. Q a = 86.243 + 2.054 - 60.238 - 4(3.369 + 3-71 1) = 24.5 19- 

... g,= 60.238 -63.311 -4(3.877 + 4.159) = -7.091- 
& = /.-£- Mq h - G{q k - g t ) + AW C ; 



INFLUENCE OF THE CYLINDER WALLS. Z l 7 

•*. Qc — 63.3 II — 2-041 — 0.0742 x 109.3 

- L973(56.35 - 21.01) + 1(1.836+ 1.847) 
= — I4.72I. 

Q d = I 9 -I + AW d ; 

••• Qd— 2.041 — 2.054 + ^(0.0299 + 0.1104) = 0.157. 
Qe=Q a + Q3 + Qc+Q d = 2.764. 

Also, equation (292) for this case gives 

Q e =Q-Mq k -G{q k -qZ-AW 

= 86.243 — 8. 1 10 — 69.723 —(3.540+4.018— 1. 841 —0.070) 
= 86.243 — 8.1 10 — 69.723 — 5.647 = 2.764. 

It is to be remembered that the heat lost by radiation and 
conduction per stroke, when estimated in this manner, is 
affected by the accumulated errors of observation and com- 
putation, which may be a large part of the total value of Q e . 

Dropping superfluous significant figures, we have in 
B. T. U. 

Q = S6.2, Q a = 24.5, Qt=- 7 .i f 

Q c =- 14-7, Qd = .06, Q e = 2.8. 

The horse-power of the engine is 

778 X 5-647 X 3692 JO 

60 x 33000 JJ 

and the steam per horse-power per hour is 

548 



16.35 



= 33-5 pounds. 



The data and results for this test and for four others made 
at the same time are given in Table II. 



3i» 



THERMODYNAMICS OF THE STEAM-ENGINE. 



W 
►J 

< 



73 
H 

h 
w 

D 

U 

< 
en 

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55 

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73 



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73 
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55 

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55 
04 



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nti « ton 




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ro o ->*-oo -*- 


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■* M f» O -<S- 
m ro m n io 


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m ro >-■ N ro 


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*j o: 



'i3MOd-3SJOq 

psjeotpuj 



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PUB UOUBipE-JJ 



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vo oo oo m vo 



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Suunp sjp3A\. c^J 5 
Aq paqjosqy 



•jsnnqxa 
Suunp S|{bav q^ 



VO O l>« o -■»- 



UOTSUBdX9 

SuunpsjiBM (^ 
Aq pappiA 



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J§UUtipS[[EM Qvf 

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jo spunod Qvj 

M Ul JB9H 



O 03 ^ 



S^ 



w 



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H 



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a^ 



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5 03 

O 03 



rt 



u 
w 

W 
CJ 
W 

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W 

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uaqcan^vj 



ti >■ b N 



00 ■*• OVO 

■>*• M O M >- 
C4 H M M 







o o o o o 

o'o'o'oo 



moco < N 

t^ N VO ■*• •*■ 

t^OO 00 00 00 

H W H M M 



t^ C4 ■<*- vo 
h p) in ro ro 
00 00 00 00 00 



•**■ ro ■*■ O O 
OO 00 vo « IO 
cj r^ to O h 

ro ro ro ro -*• 



M 00 N t~> 

h « pi moo 



ro ro ro ro ro 



O-.VO r<lNH 
r-~oo ro m m 
O rovo w t~ 

m m « ro 



ro ■*■ h ro 
m m hi N ro 



t^vo ■<*■ O 
S ■*• ro •* ro 



M ro O vo ro 
N ►> o m M 



o o o o 



O N O >n o 
O W vr O P* 



M vovo 



iH©I09"*«O 



INFLUENCE OF THE CYLINDER WALLS. 3 X 9 

Effect of Varying Cut-off. — An inspection of the inter- 
changes of heat show that the values of Q a , the heat absorbed 
by the walls during admission, increases regularly as the cut- 
off is lengthened, and that the heat returned during expansion 
decreases at the same time so that there is a considerable 
increase in the value of the heat Q c which is rejected during 
exhaust. Nevertheless there is a large gain in economy from 
restricting the cut-off so that it shall not come earlier than 
one-third stroke. Unfortunately tests on this engine with 
longer cut-off than one-third stroke have not been made, and 
consequently the poorer economy for long cut-off cannot be 
shown for this engine as for the engine of the Michigan. 

Hallauer's Tests. — In Table III are given the results of 
a number of tests made by Hallauer on two engines, one 
built by Hirn having four flat gridiron valves, and the other 
a Corliss engine having a steam-jacket. Two tests were made 
on the former with saturated steam and six with superheated 
steam. Three tests were made on the latter with satu- 
rated steam and with steam supplied to the jackets. These 
tests have a historic interest, for though not the first to which 
Hirn's analysis was applied, they are the most widely known, 
and brought about the acceptance of his method. They have 
also a great intrinsic value, as they exhibit the action of two 
different methods of ameliorating the effect of the action of 
the cylinder walls, namely, by the use of superheated steam 
and of the steam-jacket. In all these tests there was little 
compression, and Q d , the interchange of heat during compres- 
sion, is ignored. 

Superheated Steam. — Steam from a boiler is usually 
slightly moist, x, the quality, being commonly 0.98 or 0.99. 
Some boilers, such as vertical boilers with tubes through the 
steam space, give steam which is somewhat superheated, that 
is, the steam has a temperature higher than that of saturated 
steam at the boiler-pressure. Strongly superheated steam is 
commonly obtained by passing moist steam from a boiler 



320 



THERMODYNAMICS OF THE STEAM-ENGINE. 



W 

PQ 
< 





r^ 




00 




M 




^ 


w 


> 


£ 


M* 


MH 


X 


o 


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£ 


o 


w 


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175 


<-> 


CO 


*» 


HH 


<i 


H-l 


5 


Pi 


s 


o 


* 



Q 


*n 


£ 


. 


< 


.<& 




<0 


£ 

g 


^ 


H-t 


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CO 


«* 


u 


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ffi 




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pajoaCaa -_q •£ -g 




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r^ r^-o oo co ■ 


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Oj 1 MM M .MMM. MMM* 


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S5 





INFLUENCE OF THE CYLINDER WALLS. 321 

through a coil of pipe, or a system of piping, which is exposed 
to hot gases beyond the boiler. 

Superheated steam may yield a considerable amount of 
heat before it begins to condense; consequently where super- 
heated steam is used in an engine a portion of the heat 
absorbed by the walls during admission is supplied by the 
superheat of the steam and less condensation of steam occurs. 
This is very evident in Dixwell's tests given by Table XXVII, 
on page 371, where the water in the cylinder at cut-off is 
reduced from 52.2 percent to 27.4 percent, when the cut-off 
is two-tenths of the stroke, by the use of superheated steam; 
with longer cut-off the effect is even greater. This reduction 
of condensation is accompanied by a very marked gain in 
economy. 

The way in which superheated steam diminishes the action 
of the cylinder-walls and improves the economy of the engine 
is made clear by Hallauer's tests in Table III. A comparison 
of tests 1 and 3, having six expansions, shows that the heat 
Q a absorbed during admission is reduced from 28.3 to 22.4 
per cent of the total heat supplied, and that the exhaust waste 
is correspondingly reduced from 21.6 to 12.5 per cent. A 
similar comparison of tests 2 and 5, having nearly four expan- 
sions, shows even more reduction of the action of the cylinder- 
walls. The effect on the restoration of heat Q b during 
expansion appears to be contradictory: in one case there is 
more and in the other case less. It does not appear profitable 
to speculate on the meaning of this discrepancy, as it may be 
in part due to errors and is certainly affected by the unequal 
degree of superheating in tests 3 and 5. It may be noted 
that the actual value of Q c in calories is nearly the same for 
tests 1 and 2, there being a small apparent increase with the 
increase of cut-off, which is, however, less than the probable 
error of the tests. The exhaust waste Q c is much more 
irregular for tests 3 to 7 for superheated steam. The in- 
crease from 81 to 87 B. T. U. from test 6 to test 7 may 
properly be attributed to a less degree of superheating; the 



322 THERMODYNAMICS OF THE STEAM-ENGINE. 

increase from 66 to 81 B. T. U. for tests 5 and 6 is due to 
longer cut-off and less superheating; finally, the steady reduc- 
tion from 75 to 66 B. T. u. for the three tests 3, 4, and 5 is 
probably due to the rise of temperature of the superheated 
steam, which more than compensates for the effect of 
lengthening the cut-off. Finally, in test 8 the exhaust waste 
is practically reduced to zero by the use of strongly super- 
heated steam in a non-condensing engine; this shows clearly 
that the exhaust waste Q c by itself is no criterion of the value 
of a certain method of using steam. 

Steam-jackets. — If the walls of the cylinder of a steam- 
engine are made double, and if the space between the walls is 
filled with steam, the cylinder is said to be steam-jacketed. 
Both barrel and heads may be jacketed, or the barrel only 
may have a jacket; less frequently the heads only are 
jacketed. The principal effect of a steam-jacket is to supply 
heat during the vaporization of any water which may be con- 
densed on the cylinder-walls. The consequence is that more 
heat is returned to the steam during expansion and the walls 
are hotter at the end of exhaust than would be the case for 
an unjacketed engine. This is evident from a comparison of 
tests 1 and 1 1 in Table III. In test 1 only a small part of the 
heat absorbed during admission is returned during expansion, 
and by far the larger part is wasted during exhaust. In test 
1 1 the heat returned during expansion is equal to two-thirds 
that absorbed during admission, though a part of this heat of 
course comes from the jacket. About half as much is wasted 
during exhaust as is absorbed during admission. The con- 
densation of steam is thus reduced indirectly; that is, the 
chilling of the cylinder during expansion, and especially during 
exhaust, is in part prevented by the jacket, and consequently 
there is less initial condensation and less exhaust waste, and 
in general a gain in economy. But though there are many 
tests of engines both with and without steam in the jackets, 
there is so much diversity of results that engineers are not 
agreed as to the conditions under which it is profitable to use 



INFLUENCE OF THE CYLINDER WALLS. 323 

a steam-jacket, nor as the amount of gain to be expected 
under a given set of conditions. 

The jacket on the cylinder of a simple engine is usually 
supplied with steam from the main supply-pipe, so that the 
pressure in the jacket is little if any more than that in the 
cylinder during admission. As the steam in the cylinder 
comes in direct contact with the inner surface of the walls, 
while the jacket steam is in contact with the outer surface of 
the walls, the heat absorbed by the walls during admission 
comes mainly if not entirely from the steam in the cylinder. 
It is probable that the action is much the same when the 
pressure in the jackets is higher than that of the steam which 
is supplied to the cylinder, as is the case when full boiler- 
pressure is let into the jacket of the low-pressure cylinder of 
a compound engine. Now the heat supplied during expan- 
sion, though it does some work, is first subjected to a loss of 
temperature in passing from the steam in the jacket to the 
cooler water on the walls of the cylinder, and such a non- 
reversible process is necessarily accompanied by a loss of 
efficiency. On the other hand, the heat supplied by a jacket 
during exhaust passes with the steam directly into the 
exhaust-pipe. It appears, then, that the direct effect of a 
steam-jacket is to waste heat; the indirect effect (drying 
and warming the cylinder) reduces the initial condensation 
and the exhaust waste and often gives a notable gain in 
economy. 

Application to Multiple-expansion Engines. — The ap- 
plication of Hirn's analysis to the high-pressure cylinder of a 
compound or multiple-expansion' engine may be made by 
using equations (279), (280), and (282) for calculating Q a> Q b , 
and Q d , while equation (294) may be used to find Q c . 

A similar set of equations may be written for the next 
cylinder, whether it be the low-pressure cylinder of a com- 
pound engine or the intermediate cylinder of a triple engine, 
provided we can determine the value of Q ', the heat supplied 
to that cylinder. But of the heat supplied to the high-pres- 



324 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table IV. 

APPLICATION OF HIRN'S ANALYSIS TO THE EXPERIMENTAL 
ENGINE AT THE MASSACHUSETTS INSTITUTE OF TECH- 
NOLOGY. 

compound; cylinder diameters, 9 and 24 inches; stroke, 30 inches. 

Technology Quarterly , vol. XI, p. 43. 



Test numbers. 



Duration of test, minutes 

Total number of revolutions 

Revolutions per minute 

Steam-consumption during test, pounds: 

Passing- through cylinders 

Condensation in h. p. jacket 

" in receiver jacket 

" in 1. p. jacket 

Total 

Condensing water for test, pounds 

Priming, by calorimeter 

Temperatures, Fahrenheit: 

Condensed steam 

Condensing water, cold 

Condensing water, hot 

Pressure of the atmosphere, by the barometer, 

pounds per square inch . . . . 

Boiler-pressure, lbs. per sq. in., absolute 

Vacuum in condenser, inches of mercury 

Events of the stroke, per cent: 
High-pressure cylinder — 

Cut-off, crank end 

head end 

Release, both ends 

Compression, crank end 

head end 

Low-pressure cylinder — 

Cut-off, crank end 

head end 

Release, both ends 

Compression, crank end. 

head end 

Absolute pressures in the cylinder, pounds 
per square inch: 
High-pressure cylinder — 

Cut-off, crank end 

head end 

Release, crank end 

headend 

Compression, crank end 

head end 

Admission, crank end 

head end 

Low-pressure cylinder — 

Cut-off, crank end '. 

headend . . 

Release, crank end 

head end 

Compression, crank end 

head end ... 

Admission, crank end 

head end 

Heat-equivalents of external work, b. t. u. 
from areas on indicator-diagram to line of 
absolute vacuum: 

High-pressure cylinder — 
During admission, A Wa, crank end. 
head end. 



I. 



60 
4976 
82.93 

844-5 
42.0 

37-5 
82.8 



1006.8 
16066 



93-4 

52.2 

105.8 

14-5 

114. 7 

26.6 



16.8 

15-7 
100. o 

7-5 
"•5 

18.0 

18.0 

100. o 

3-5 

5-5 



106.5 
107. 1 
29.9 
24.7 
17.2 
17.8 

32.7 
42.2 

14.4 
13.6 
3-4 
3-3 
1.9 
2.4 

3-4 
4.4 



3-59 
3-49 



II. 



60 

4974 
82.90 

820.0 
39-° 
5i-5 
74-5 



16287 



91.4 

52.2 

104.2 

14-5 

"5-5 

26.6 



15-8 

15.4 

100. o 

7-5 

"•5 

18.0 

18.0 

100. o 

3-5 

5-5 



109 
107 
29 
24 
17 
19 
35 
42 

14 

13 
3 

3 



3-38 
3-47 



III. 



60 

4903 
81 .72 

1059.0 

42-5 

73-o 
88.5 



1263.0 

2 1 800 
i- x 3 

86.0 

52.5 
102.6 

14-5 

"5-4 

26.3 



23.8 

24.2 

100. o 

7-5 
10.5 

24-5 

25.0 

100. o 

3-5 
4.0 



107.5 
107.5 
37-o 
33-2 
18.0 
20. 1 
36.5 
45-2 

14.7 

14.2 

4.2 

4.4 

2.2 



2.8 

4.1 
5-8 



513 

5-43 



IV. 



60 
4882 

81.37 

1086.0 

44.0 
80.0 



21960 
0.9* 



52-7 

104.0 

14.5 

"5-9 

26.3 



25.2 

25.6 

100. o 

7-5 

10.5 

24.5 

25.0 

100. o 

3-5 
4.0 



108.3 

108. 1 

38.2 

33-7 
18.7 
21.3 
38.2 
46.9 

151 

14.9 
42 

4-5 
2.2 



5-45 
5-8 3 



INFLUENCE OF THE CYLINDER WALLS. 

Table IV — Continued. 



325 



Test numbers. 


I. 


II. 


IV. 


IV. 


High-pressure cylinder — 










During expansion, A Wb, crank end 


7-94 


8.05 


8.38 


8-43 


head end 


7.81 


7-97 


8.48 


8-57 




2.86 


2.96 


3- 11 


3.28 




2.91 


3-°3 


3-i8 


3-54 


During compression, A JVd, crank end. 


o.35 


o.37 


0.38 


0.40 


head end.. 


0.67 


0.72 


0.72 


0.72 


Low-pressure cylinder — 










During admission, A IVa, crank end.. . . 


4-13 


4.15 


5-93 


6. 11 


head end .... 


4.07 


4-13 


5-99 


6.25 


During expansion, A Wb, crank end. . . . 


7.19 


7-25 


7.78 


7-99 


head end 


7-°5 


7.12 


7.88 


8.16 




2.64 


2-55 


2.87 


2.98 




3-25 


3.10 


3-98 


4.00 


During compression, A Wd, crank end.. 


0.12 


0.13 


0.17 


0.17 


head end . . 


o.33 


0.32 


0.28 


0.29 


■Quality of the steam in the cylinder. At ad- 










mission and at compression the steam was 










assumed to be dry and saturated: 










High-pressure cylinder- 












70.18 
83.98 


71. 11 
85-59 


73.60 
84.97 


76.24 
83-73 




Low-pressure cylinder — 


At cut-off Xi 


91.94 
87.62 


95.20 
88.19 


92.13 
88.57 


92.58 
87.86 




Interchanges of heat between the steam and 










the walls of the cylinders, in b. t. u. 










Quantities affected by the positive sign are 










absorbed by the cylinder- walls; quantities 










affected by the negative sign are yielded 










by the walls: 










High-pressure cylinder — 












99.82 

25.22 

— 15.01 


97.10 

23-85 
-15.84 


126.90 

27.14 

— 16.13 


131.30 

25.83 
— 12.60 


During admission Qa 






— 11.13 
0.98 
3-7° 


— 10.04 
o-45 
3-45 


-12.99 
0.46 
3-8i 


— 15-47 
0.47 

3-97 








1.87 


1.87 


2 20 


2.20 


Second intermediate receiver — 












3-3° 


4-53 


6.50 


7.19 




o.6q 


0.69 


1.60 


1.60 


Low-pressure cylinder — 












96.24 


94.26 


123.40 


128.53 


During admission Q'a 


7.63 

— i-75 
— 11. 11 


5-44 


9-63 


io-57 

— 0.97 

-16.53 






— 7-53 


— 9-92 




o-33 
7.30 
4.44 






0.64 
7-85 
5-70 


Supplied by jacket . . . . Q'j 


6.56 
4-44 


7.91 
5.70 




Total loss by radiation: 




7-39 
7.00 


7-39 
5-78 


9-52 
7-83 


9-52 
4.42 


By equation . . . . 


Power and economy: 


Heat-equivalents of works per stroke: 












8 02 


7.89 
8.27 




10.17 
10.53 




8.10 


10.04 


Totals 


16.12 
14.30 


16.16 
14-54 


20.05 


20.70 
19.01 


Total heat furnished by jackets 


Distribution of work: 






1. 00 


1. 00 


1 .00 


1. 00 


Low-pressure cylinder 




1.05 

63.19 
15-59 




1 .04 


Total horse-power 


63.07 
15.96 


78.13 
16.16 


80.47 
16.14 


Steam per horse-power per hour 


B. T. U. per horse-power per minute — 


282.1 


275- 1 


284.4 


283 8 



326 THERMO D YNAMICS OF THE STEAM-ENGINE. 

sure cylinder a part is changed into work, a part is radiated,, 
and a part is rejected in the exhaust waste. The heat rejected 
is represented by 

Q+Q,-AW-Q„ (295) 

where Q is the heat supplied by the steam entering the 
cylinder, Q y is the heat supplied by the jacket, AW Is the 
heat-equivalent of the work done in the cylinder, and Q e is 
the heat radiated. Suppose the steam from the high-pressure 
cylinder passes to an intermediate receiver, which by means 
of a tubular reheater or by other means supplies the heat Q r1 
while there is an external radiation Q re% The heat supplied 
to the next cylinder is consequently 

Q , = Q+Q J -AW-Q,+ Q r -Q„. . . (296) 

In a like manner we may find the heat Q" supplied to the 
next cylinder; for example, to the low-pressure cylinder of a 
triple engine. 

It is clear that such an application of Hirn's analysis can 
be made only when the several steam-jackets on the high- and 
the low-pressure cylinders, and the reheater of the receiver, 
etc., can be drained separately, so that the heat supplied ta 
each may be determined individually. 

Table IV gives applications of Hirn's analysis to four tests 
on the experimental engine in the laboratory of the Massa- 
chusetts Institute of Technology, using the high- and low- 
pressure cylinders only as a compound engine. 

Table V gives application to four tests on the same engine 
running as a triple engine. 

It will be noted that in both sets of tests the steam in the 
cylinders becomes drier in its course through the engine, 
under the influence of thorough steam-jacketing with steam 
at boiler-pressure. In the triple-expansion tests the steam is 
practically dry at release in the low-pressure cylinder. All of 
the tests show superheating in the low-pressure cylinder, 
which is of course possible, for the steam in the jackets is at 



INFLUENCE OF THE CYLINDER WALLS. 

Table V. 



327 



APPLICATION OF HIRN'S ANALYSIS TO THE EXPERIMENTAL 
ENGINE IN THE LABORATORY OF THE MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY. 

triple-expansion; cylinder diameters, 9, 16, and 24 inches ; 
stroke, 30 inches. 
Trans. Am. Soc. Mech. Engrs., vol. XII, p. 740. 





I. 


II. 


III. 


IV. 




60 


60 


60 


60 




5299 


5228 


5173 


5148 




88.3 


87.1 


86.2 


85.8 


Steam-consump'n during test, lbs. : 












1 193 


ii57 


1234 


1305 


Condensation in h. p. jacket.. 


57 


50 


29 


30 


" in first receiver- jacket 


61 


64 


69 


72 




85 


92 


97 


I05 


" in sec'd receiver-jacket 


53 


50 


52 


51 




89 


76 


90 


87 


Total 


1538 
22847 


1489 
22186 


i57i 
20244 


16^0 


Condensing water for test, lbs. . . 


20252 




0.013 


0.012 


. 01 1 


012 


Temperatures, Fahrenheit: 












95-4 
41.9 


92. 1 
42. 1 


102.4 
43-0 


105-3 
42.8 






96.1 


96.6 


106.3 


IO9.6 


Pressure of the atmosphere, by the 












14.8 


14.8 


14.7 


I4.7 


Boiler pressure, lbs. per sq. in. 












155.3 


155-5 


156.9 


157.7 


Vacuum in condenser, inches of 




25.0 


25.1 


24. 1 


23.9 


Events of the stroke : 


High-pressure cylinder — 












0.192 


0.194 


0.245 


o.:83 




0.215 


0.205 


0.271 


o.;o5 




1. 00 


1. 00 


1. 00 


1. 00 


Compression, crank end.. 


0.05 


0.05 


0.04 


0.04 


head end. . . 


0.05 


0.05 


0.05 


0.05 


Intermediate cylinder — 












0.29 


0.29 


0.29 


29 




1. 00 


1. 00 


1. 00 


1. 00 


Compression, crank end.. 


0.03 


0.03 


03 


0.03 


head end.. . . 


0.04 


0.04 


0.04 


0.04 


Low-pressure cylinder — 












0.38 


0.38 


0.38 


0.38 




o.39 


o.39 


o.39 


0.39 




1. 00 


1. 00 


1. 00 


T .00 



328 THERMODYNAMICS OF THE STEAM-ENGINE. 

TABLE V — Continued. 



Absolute pressures in the cylin- 
der, pounds per sq. in.: 
High- pressure cylinder — 

Cut-off, crank end 

head end 

Release, crank end 

head end 

Compression, crank end... 
head end.. . . 

Admission, crank end 

head end 

Intermediate cylinder — 

Cut-off, crank end 

head end 

Release, crank end 

head end 

Compression, crank end... 
head end.... 

Admission, crank end 

head end 

Low-pressure cylinder — 

Cut-off, crank end 

head end 

Release, crank end 

head end 

Compression and admission — 

crank end 

head end 

Heat-equivalents of external work, 
B. T. U. , from areas on indica- 
tor-diagram to line of absolute 
vacuum: 

High-pressure cylinder — 
During admission, 

A W a , crank end 

head end 

During expansion, 

A Wb, crank end 

head end 

During exhaust, 

A W c , crank end 

head end 

During compression, 

A Wd, crank end 

head end 

Intermediate cylinder — 
During admission, 

A Wd , crank end 

head end 

During expansion, 

A Wb ', crank end 

head end 



145 

143 
41 
41 
43 
48.7 
64 -5 
75-3 



37-2 
35.o 
13-6 
13-4 
16.3 

17.9 
20.4 
21. 1 

12. 1 

12.0 

5.6 

5-4 

3-7 
4-3 



II. 



5.7i 
6.61 

10.65 
10.81 

7-73 
8.08 

0.48 
0.62 



7.58 
7-43 

9-54 
9.22 



145 
143 

4i 

40 

45 

47-9 
68.8 

74.8 



37-6 

35-3 
14 2 

13.8 

17-3 
18.8 
20.8 

22.8 

12.6 
12.4 

5-3 
5-8 

3.8 
4-5 



III. 



5.78 
6-37 

10.76 
11.04 

7.89 
8.15 

0.60 
0.64 



7-57 
7-55 

9-54 
9-3i 



138.8 
140.3 

44-7 

45-7 
48.5 
54-5 
72.2 
86.7 

38.6 
39-6 
14.7 

14.9 
18.2 
20.3 
22.2 
24.2 

12.4 

I3-I 
5-i 
5-9 

4.1 
4.6 



IV. 



7.00 
8.42 

10.40 
11.22 

8.44 
9.04 

0.49 
o-73 



7.98 
8.46 

9.91 
10.37 



138.3 
140.6 

48.4 
49.8 
53-2 
62.0 
81.2 
97.8 

40.9 
42.6 
16.0 
16.0 
19.0 
22.4 
23.1 
26.7 

13.2 
14.0 

5-7 

6.4 

4.2 
4-7 



8.19 

9-50 

10.25 
11.09 

9.02 
9.66 

0.50 
0.81 



8.64 
9.10 

10.64 
11. 14 



INFLUENCE OF THE CYLINDER WALLS. 

Table V — Continued, 



329 





I. 


II. 


III. 


IV. 


Intermediate cylinder — 










During exhaust, 












9.27 


9-47 


9.64 


10.54 


head end 


9.27 


9-47 


10.18 


10.84 


During compression, 






o.39 


0-43 


0.57 


0.46 




0.60 


0.70 


0.78 


0.84 


Low-pressure cylinder — 










During admission, 












7-75 


7-95 


8-33 


8.97 




7-99 


8.19 


8.66 


9-39 


During expansion, 












6.83 


7.10 


6.86 


7-45 




6.87 


7.12 


7-34 


7.87 


During exhaust, 












5-o8 


5.08 


4.62 


5.09 




5.08 


5.16 


4.81 


5-oo 


During compression, 












0.00 


0.00 


0.00 


0.00 




0.00 


0.00 


0.00 


0.00 - 


Quality of the steam in the cylin- 










der. At admission and at com- 










pression the steam was assumed 










to be dry and saturated : 










High-pressure cylinder — 












0.785 


0.784 


0.848 


0.875 




0.899 


0.903 


0.920 


0.931 


Intermediate cylinder — 


At cut-off . . .xi'. . 


0.899 


0.912 


0.906 


0.908 


At release xj '. . 


0.994 


superheated. 


superheated. 


superheated. 


Low-pressure cylinder — 




0.978 


superheated. 


O.970 


O.974 


At release X?" . 


superheated. 


superheated. 


superheated. 


superheated. 


Interchanges of heat between the 


steam and the walls of the cylin- 










ders, in b. t. u. Quantities 










affected by the positive sign are 










absorbed by the cylinder-walls; 










quantities affected by the nega- 










tive sign are yielded by the 










walls : 


1 








High-pressure cylinder — 










Brought in by steam . . Q. 


132.93 


130.77 


141. II 


149.84 




23.54 


23-43 


17.49 


14-93 


During expansion Q^.. 


— 18.69 


— 19.28 


- 15-33 


- 1403 


During exhaust Q c . . 


- 8.36 


— 7.22 


-3 50 


- 2.38 


During compression . . . Qd . 


O.45 


0.51 


0.49 


O.52 


Supplied by jacket . . . ,Qj.. 


4-56 


4.08 


2-39 


2.50 


Lost by radiation Q e .. 


I.50 


1.52 


1-54 


1.54 


First intermediate receiver — 










Supplied by jacket . . . .Q r . 


4.92 


5.20 


5.67 


5-95 


Lost by radiation.. . . . ,Q re . 


O.58 


0.58 


o.59 


0.59 



33Q 



THERMODYNAMICS OF THE STEAM-ENGINE. 

Table V — Continued. 





I. 


II. 


III. 


IV. 


Intermediate cylinder — 










Brought in by steam . . Q'. . 


131.89 


129.61 


137-87 


146.64 


During admission. . . . (V '. 


13.62 


11.74 


n-33 


11.75 




- 18.65 


— 18.84 


— 20.30 


— 21.88 


During exhaust Qc' . 


0.22 


1-57 


2.88 


3.41 


During compression . . . Q<j . 


0.44 


0.51 


0.62 


o.59 


Supplied by jacket. . . . Qf . 


6.82 


7.50 


7.97 


8.64 




2.45 


2.48 


2.50 


2.51 


Sec'd intermediate receiver — 










Supplied by jacket .... Q r ' . 


4.20 


4.04 


4.27 


4.22 




1.20 


1.22 


1.23 


1.24 


Low-pressure cylinder — 










Brought in by steam . . Q" . 


132.14 


130-50 


138.61 


147.33 




5.85 


3-05 


5.57 


5.29 




-9-5i 


- 7.09 


-8.65 


— 10.13 




2.53 


2.23 


- 1.44 


— 0.1 1 


During compression. . . Qd" 


0.00 


0.00 


0.00 


0.00 


Supplied by jacket .... Q/' 


7.08 


6.20 


7.41 


7.14 




4-34 


4.40 


4-45 


4-47 


Total loss by radiation — 










By preliminary tests, 2Q e . 


10.07 


10.20 


10.31 


10.35 




11.68 


10.19 


8-75 


8.07 


Power and economy: 










Heat-equivalents of works 










per stroke — 










H. P. cvlinder AW.. 


8.44 


8-34 


9.17 


9' 52 


Interm. cylinder. . ..A W'.. 


7.12 


6-95 


7-77 


8.42 


L. P. cylinder A W". 


9.64 


10.06 


10.87 


11.79 


Totals 


25.20 


25.35 


27.81 


29- 73 


Tot. heat furnished by jackets 


27.58 


27.02 


27.71 


28.45 


Distribution of work — 










High-pressure cylinder.. . . 


1. 00 


1. 00 


1. 00 


1. 00 




0.84 


0.83 


0.85 


0.88 


Low-pressure cylinder 


1. 14 


1. 21 


1. 19 


1.24 




104.9 
14.65 


104.2 
14.31 


113. 1 
13.90 


120.3 

13-73 


Steam per H. P. per hour. . . 


B. T. U. per H. P. per minute 


247 


241 


236 


232 



full boiler-pressure while the steam in the cylinder is below 
atmospheric pressure. The superheating was small in all 
caS es — not more than would be accounted for by the errors 
of the tests. The exhaust waste Q c " from the low-pressure 
cylinder in the triple-expansion tests is very small in all cases 
— less than two per cent of the heat supplied to the cylinders. 
The apparent absurdity of a positive value for Q e " in two of 



INFLUENCE OF THE CYLINDER WALLS. 331 

the tests (indicating an absorption of heat by the cylinder 
walls during exhaust) may properly be attributed to the 
unavoidable errors of the test. 

In the fourth test, when the engine was developing 120.3 
horse-power, there were 1305 pounds of steam supplied to the 
cylinders in an hour, and 345 pounds to the steam-jackets; so 
that the steam per horse-power per hour passing through the 
cylinders was 

1305 -7- 120.3 = 10.86 pounds, 

while the condensation in the jackets was 

345 -r- 120.3 — 2 -%7 pounds. 

So that, as shown on page 245, the B. T. u. per horse-power 
per minute supplied to the cylinders by the entering steam 
was 191. 1, while the jackets supplied 40.6 B. T. u., making 
in all 231.7 B. T. U. per horse-power per minute for the heat- 
consumption of the engine. In the same connection it was 
shown that the thermal efficiency of the engine for this test 
was 0.183, while the efficiency for incomplete expansion in a 
non-conducting cylinder corresponding to the conditions of 
the test was 0.222 ; so that the engine was running with 0.824 
of the possible efficiency. In light of this satisfactory con- 
clusion some facts with regard to the test are interesting. 

It will be noted that 149.84 B. T. U. per stroke are brought 
in by the steam supplied to the high-pressure cylinder and 
that 28.45 B - T « u - P er stroke are supplied by the steam- 
jackets; and that, further, 29.73 B. T. U. are changed into work 
while 10.35 are radiated. Thus it appears that the jackets 
furnished almost as much heat as was required to do all the work 
developed. Of the heat furnished by the jackets something 
more than a third was radiated; the other two-thirds may 
fairly be considered to have been changed into work, since 
the exhaust waste of the low-pressure cylinder was practically 
zero. 

Table VI gives the results of tests made on a vertical 



33 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

Table VI. 

TRIPLE-EXPANSION VERTICAL PUMPING-ENGINE AT 
MILWAUKEE, WISCONSIN. 

CYLINDER DIAMETERS, 28, 48, AND 74 INCHES J STROKE, 60 INCHES. 

By Professor R. C. Carpenter, Trans. Am. Soc. Mech. Engrs., 

vol. xv, p. 313. 

Duration, hours , 24 

Revolutions per minute 20.3 

Pressures, pounds per square inch : 

Gauge at throttle 121. 5 

Vacuum •. 13.8 

Barometric 14.5 

High-pressure jackets 121. 4 

Intermediate and low-pressure jackets 56.5 

First receiver 32.4 

Second receiver. 13.8 

Moisture in steam, per cent 1.1 

Steam used in jackets, per cent 9.3 

Indicated horse-power , 573-9 

Pump, horse-power 521.0 

Friction, per cent g,2 

Steam per horse-power per hour H.8 

B. T. U. per horse-power per minute 217.6 

Hirn's analysis, interchanges of heat between the walls of the 
cylinder and the steam ; in per cent of total heat per stroke : 
High-pressure cylinder — 

Absorbed during admission , 10.5 

Restored during expansion — 7.0 

Rejected during exhaust „ : — 4.7 

Absorbed during compression 0.2 

Intermediate cylinder — 

Absorbed during admission n.8 

Restored during expansion . — 6.2 

Rejected during exhaust . —7.2 

Given up during compression — 0.2 

Low-pressure cylinder — 

Absorbed during admission. 2.1 

Restored during expansion — 0.1 

Rejected during exhaust , — 21.2 

Given up during compression — 1.2 

Thermodynamic efficiency 0.194 



INFLUENCE OF THE CYLINDER WALLS. 333 

triple-expansion pumping-engine, together with an application 
of Hirn's analysis, which is especially interesting as the engine 
was made by the builders of the experimental engines at the 
Massachusetts Institute of Technology, and it ran under 
much the same conditions. Its higher efficiency and better 
economy was due in part to the fact that the engine was 
larger and superior in certain details, but more particularly to 
the good vacuum attained. The principal difference in the 
results of the application of Hirn's analysis to the tests on the 
two engines is in the heat rejected by the walls during 
exhaust; for the experimental engine these quantities are 
small for all three cylinders, and in particular is practically 
zero for the low-pressure cylinder, while for the pumping- 
engine they are of considerable magnitude, and the exhaust 
waste for the low-pressure cylinder is large. So far as any 
significance can be attached to this comparison it shows that 
a considerable exhaust waste is not incompatible with a high 
economy. 

Alternate Method for Compound Engines. — When the 
method of applying Hirn's analysis to a compound engine, 
which has just been explained and exemplified, can be used 
it is most satisfactory, as the comparison of the observed and 
calculated radiation losses gives an excellent check on the 
accuracy of the test in general. If, however, it should be 
impossible in any case to properly account for the heat added 
to or lost by the steam in the intermediate receiver, we may 
still make an application of Hirn's analysis by the following 
method. 

There is no difficulty in making the application to the 
high-pressure cylinder by the method already explained. 
The trouble comes in the determination of the heat Q sup- 
plied by the steam which enters the low-pressure cylinder. 
But by applying equation (292) to the low-pressure cylinder 
we have 

QI = Q'+QJ-Mq- Gig>-qi)-A( JV.+ W t - W c - W d ), (297) 



334 THERMODYNAMICS OF THE STEAM-ENGINE. 

and consequently 

a = Q:-Q/+^ t +G(^- ?l )+A(w a +m-w-w d ), ( 29 8) 

by aid of which equation Q may be eliminated from the equa- 
tion for Q a ', the heat absorbed by the wall of the low-pressure 
cylinder during admission. Therefore we have for the equa- 
tions for a compound engine 

Q a = Q + I -I l -AlV a ; (299) 

Q i = I 1 -I i -AW i ; (300) 

& = I.-A-Q-Qi+Q.+ A(W+W e );. (301) 

Q d = f 3 -I o + AW d ; (302) 

+ A{W>- W c - JV d ); . . . (303) 

ot = i;-i;-Awi\ (304) 

QJ = I i >--I 9 >--M<? i -G(<? k -<? t ) + AW e ' ] . (305) 

Q d ' = i:-i:+Aw d f (306) 

Maire's Tests. — In Table VII is given a large number 
of tests on important engines of various types, together with 
the application of Hirn's analysis. Some of the engines 
tested, such as the Cornish engine and the Bull engine, are 
out of date, and the interest in some others is impaired by 
the low pressure of steam in the boiler. It is also unfor- 
tunate that Maire did not separate the interchanges of heat 
during the admission and expansion in the low-pressure 
cylinder, nor does his report allow us to make the separation. 
We have consequently for the low-pressure cylinder only the 
exhaust waste Q c and the exchange Q d for compression. 
Nevertheless the table is instructive and important. 



INFLUENCE OF THE CYLINDER WALLS. 335 

The dimensions of the engines and the conditions of the 
tests were as follows: 

The steam-consumption was determined by measuring the 
feed-water supplied to the boiler. The steam condensed in 
the steam-jackets was collected and weighed separately. The 
air-pump discharge was allowed to flow through an orifice 
under a measured head ; the coefficient of discharge for the 
orifice was determined by direct experiments. The per cent 
of priming in the steam was determined by calorimetric tests. 

The test I was made on a single-cylinder rotative pump- 
ing-engine, having a diameter of 45 inches and a stroke of 
5 feet 6 inches. The sides and ends of the cylinder were 
jacketed with boiler steam. The steam was distributed by 
separate slide-valves near the ends of the cylinder, with 
expansion-plates adjustable by hand, on the backs of the 
main valves. 

The tests 2, 3, and 4 were made on a Woolf beam-engine 
driving a flour-mill. The cylinders were 24-J inches in 
diameter by 3 feet 5 inches stroke, and 38 inches diameter 
by 5 feet 6 inches stroke, and were steam-jacketed on the 
sides only. The steam was distributed to the high-pressure 
cylinder by a slide-valve with cut-off plates on the back, and 
to the low-pressure cylinder by a piston-valve, all being 
worked by eccentrics. 

The tests 5 and 6 were made on an unjacketed horizontal 
Woolf engine, of a type very commonly used in factories in 
Lancashire. The cylinders were I5f and 28J- inches in 
diameter by 4 feet 3 inches stroke. The piston speed was 
about 680 feet per minute and the load was light, so that the 
steam was much wire-drawn. 

The test 7 was made on a compound beam receiver-engine 
with the cranks at right angles, working pumps directly from 
the beams. The cylinders were 21 and 36 inches in diameter, 
and the stroke was 5 feet 6 inches. Both cylinders, with the 
exception of the high-pressure cylinder and the receiver- 
covers, were jacketed with boiler steam. The steam was 



33^ 



THERMODYNAMICS OF THE STEAM-ENGINE. 



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INFLUENCE OF THE CYLINDER WALLS. 



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33^ THERMODYNAMICS OF THE STEAM-ENGINE. 

distributed by slides, one at each end of each of the cylinders, 
with cut-off plates adjustable by hand. 

The test 8 was made on a Cornish engine working a 
single-acting piston-pump direct from the beam, and having 
the usual Cornish, double-beat, steam, equilibrium, and 
exhaust-valves, a single-acting air-pump, and a jet-condenser; 
the cylinder was 68J inches in diameter by 8 feet stroke, and 
was jacketed on the sides with boiler steam. The pump 
delivered its water on the up or steam stroke, so that the 
preponderance of weight on the pump-pole was only enough 
to overcome the suction lift. The valves and piston were 
inspected to assure their tightness before the test. The 
engine was doing the highest duty at the West Middlesex 
Waterworks, and was taken as one of the best engines of its 
type now working. This type of engine was developed in 
Cornwall, where it was used to pump water from deep mines 
by a pump-rod hung directly from one end of the beam while 
the piston was hung from the other end of the beam. It had 
no fly-wheel, but the pump-rod, beam, and counter-weights 
made in the aggregate a large reciprocating mass, that 
absorbed work during the first part of the stroke when the 
steam-pressure in the cylinder was high, and restored that 
work and assisted the steam to complete the stroke, after it 
had lost pressure through expansion, during the latter part of 
the stroke. Such a reciprocating mass is essential to the 
proper action of the engine with expansion. The pumps of 
the original engines were worked by the weight of the rods 
during the return or equilibrium stroke, at which time there 
was free communication between the two ends of the cylinder. 
The lower end of the cylinder was open to the condenser 
during the steam-stroke. 

The tests g and io were made on a single-cylinder beam 
rotative pumping-engine, having a diameter of 32 inches and 
a stroke of 5 feet 6 inches. The cylinder sides and base were 
jacketed with boiler steam. Steam was distributed by slide- 



INFLUENCE OF THE CYLINDER WALLS. 339 

valves at the top and bottom of the cylinder, with cut-off 
plates, adjustable by hand, on the backs of the main valves. 

The tests II and 12 were made on a single-cylinder beam 
rotative engine, similar to the one just described, and taking 
steam from the same boilers. The cylinder was 27 inches in 
diameter by 6 feet stroke. 

The test 13 was made on a Bull engine with a cylinder 68 
inches in diameter by 10 feet stroke, driving direct a 45-inch 
plunger-pump, and forcing water to a height of 40 to 55 feet. 
The valves and gear were of the usual Cornish pattern, and 
the sides and base of the cylinder were steam-jacketed. This 
type of engine differs from the Cornish engine in not having 
a beam, and though the pump-rod is loaded there is seldom 
sufficient reciprocating mass to allow of much expansion. In 
the case of the engine tested only if expansions could be 
obtained. For convenience, the steam-stroke is detailed 
under the heading of the high-pressure cylinder, and the 
exhaust-stroke under the heading of the low-pressure cylinder. 

The tests 14 and 15 were made on a Woolf beam rotative 
engine, working a double-acting pump. The cylinders were 
29 inches diameter by 5 feet 5 inches stroke, and 47^- inches 
diameter by 8 feet stroke, and jacketed with steam on the 
sides and ends. 

The most interesting tests for our present purpose are 
2, 3, and 4, since they show how a compound engine is 
affected by steam-jackets. It appears that in this case the 
interchanges of heat for the high-pressure cylinder are not 
much affected, though there is some increase in the heat 
restored during expansion, and the exhaust waste is less when 
steam is used in the jackets. But the exhaust waste for the 
low-pressure cylinder is very much reduced, and may in a large 
measure account for the gain in economy. 

The comparison of test 7 with tests 14 and 15 is instruc- 
tive, for the latter show practically no exhaust waste from the 
low-pressure cylinder, while the former has a considerable 
exhaust waste ; nevertheless the economy for test 7 is notably 



34° THERMODYNAMICS OF THE STEAM-ENGINE. 

better than for tests 14 and 15. The advantage is probably- 
due to the higher steam-pressure and larger number of 
expansions enjoyed by the engine represented by test 7. 
This is only one more example to show that while the exhaust 
waste is always a direct loss it may be unadvisable to try to 
reduce it to zero by the lavish application of steam-jackets or 
by other ways of supplying heat to steam on its way through 
a compound or multiple-expansion engine. It will be shown 
later that it is possible to overdo the application of heat in 
this way and so injure the economy of the engine; the best 
result appears to be attained by a judicious or a fortunate 
compromise of the gain from expansion and the loss from 
condensation and evaporation, together with the amelioration 
of the latter by the use of steam-jackets and intermediate 
reheaters. 

Quality of Steam at Compression. — In all the work of 
this chapter the steam in the cylinder at compression has been 
considered to be dry and saturated, and it has been asserted 
that little if any error can arise from this assumption. It is 
clear that some justification for such an assumption is needed, 
for a relatively large weight of water in the cylinder would 
occupy a small volume and might well be found adhering to 
the cylinder walls in the form of a film or in drops; such a 
weight of water would entirely change our calculations of the 
interchanges of heat. The only valid objection to Hirn's 
analysis is directed against the assumption of dry steam at 
release. Indeed, when the analysis was first presented some 
critics asserted that the assumption of a proper amount of 
water in the cylinder is all that is required to reduce the cal- 
culated interchanges of heat to zero. It is not difficult to 
refute such an assertion from almost any set of analyses, but 
unfortunately such a refutation cannot be made to show con- 
clusively that there is little or no water in the cylinder at 
compression; in every case it will show only that there must 
be a considerable interchange of heat. . 

For the several tests on the Him engine given in Table 



INFLUENCE OF THE CYLINDER WALLS. 34 1 

III, Hallauer determined the amount of moisture in the steam 
in the exhaust-pipe, and found it to vary from 3 to 10 per 
cent. Professor Carpenter* says that the steam exhausted 
from the high-pressure cylinder of a compound engine showed 
12 to 14 per cent of moisture. Numerous tests made in the 
laboratory of the Massachusetts Institute of Technology show 
there is never a large percentage of water in exhaust-steam. 
Finally, such a conclusion is evident from ordinary observation. 
Starting from this fact and assuming that the steam in the 
cylinder at release is at least as dry as the steam in the 
exhaust-pipe, we are easily led to the conclusion that our 
assumption of dry steam is proper. Professor Carpenter 
reports also that a calorimeter test of steam drawn from the 
cylinder during compression showed little or no moisture. 
Nevertheless there would still remain some doubt whether the 
assumption of dry steam at compression is really justified, 
were we not so fortunate as to have direct experimental 
knowledge of the fluctuations of temperature in the cylinder 
walls. 

Dr. Hall's Investigations. — For the purpose of studying 
the temperatures of the cylinder-walls Dr. E. H. Hall used 
a thermo-electric couple, represented by Fig. 74. / is a cast- 



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Fig. 74- 



iron plug about three-quarters of an inch in diameter, which 
could be screwed into the hole provided for attaching an 
indicator-cock to the cylinder of a steam-engine. The 
inner end of the plug carried a thin cast-iron disk, which was 
assumed to act as a part of the cylinder-wall when the plug 
was in place. To study the temperature of the outside sur- 
face of the disk a nickel rod N was soldered to it, making a 

* Trans. Am. Soc. Mech. Engrs., vol. xil, p. 811. 



34 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

thermo-electric couple. Wires from / and N led to another 
couple made by soldering together cast-iron and nickel, and 
this second couple was placed in a bath of paraffine which 
could be maintained at any desired temperature. In the 
electric circuit formed by the wires joining the two thermo- 
electric couples there was placed a galvanometer and a circuit- 
breaker. The circuit-breaker was closed by a cam on the 
crank-shaft, which could be set to act at any point of the 
revolution. If the temperature of the outside of the disk iT 
differed from the temperature of the paraffine bath at the 
instant when contact was made by the cam, a current passed 
through the wires and was indicated by the galvanometer. 
By properly regulating the temperature of the bath, the 
current could be reduced and made to cease, and then a 
thermometer in the bath gave the temperature at the surface 
of the disk for the instant when the cam closed the electric 
circuit. Two points in the steam-cycle were chosen for 
investigation, one immediately after cut-off and the other 
immediately after compression, since they gave the means of 
investigating the heat absorbed during compression and 
admission of steam, and the heat given up during expansion 
and exhaust. 

Three different disks were used : the first one half a milli- 
metre thick, the second one millimetre thick, and a third 
two millimetres thick. From the fluctuations of temperature 
at these distances from the inside surface of the wall some 
idea could be obtained concerning the variations of tempera- 
ture at the inner surface of the cylinder, and also how far the 
heating and cooling of the walls extended. 

The account given here is intended only to show the 
general idea of the method, and does not adequately indicate 
the labor and difficulties of the investigation which involved 
many secondary investigations, such as the determination of 
the conductivity of nickel. Having shown conclusively that 
there is an energetic action of the walls of the cylinder, 
Dr. Hall was unable to continue his investigations. 



INFLUENCE OF THE CYLINDER WALLS. 343 

Callendar and Nicolson's Investigations. — A very re- 
fined and complete investigation of the temperature of the 
cylinder walls and also of the steam in the cylinder was made 
by Messrs. Callendar and Nicholson* in 1895 at the McGill 
University, by the thermoelectric method. 

The wall temperatures were determined by a thermoelec- 
tric couple of which the cylinder itself was one element and 
a wrought-iron wire was the other element. To make such a 
couple, the cylinder-wall was drilled nearly through, and the 
wire was soldered to the bottom of the hole. Eight such 
couples were established in the cylinder-head, the thickness 
of the unbroken wall varying from 0.01 of an inch to 0.64 of 
an inch. Four pairs of couples were established along the 
cylinder-barrel, one near the head, and the others at 4 inches, 
6 inches, and 12 inches from the head. One of each pair of 
wall couples was bored to within 0.04 of an inch, and the 
other to 0.5 of an inch of the inside surface of the cylinder. 
Other couples were established along the side of the cylinder 
to study the flow of heat from the head toward the crank end. 

The engine used for the investigations was a high-speed 
engine, with a balanced slide-valve controlled by a fly-wheel 
governor. During the investigations the cut-off was set at a 
fixed point and the speed was controlled externally. By the 
addition of a sufficient amount of lap to prevent the valve 
from taking steam at the crank end the engine was made 
single-acting. The normal speed of the engine was 250 revo- 
lutions per minute, but during the investigations the speed 
was from 40 to 90 revolutions per minute. The diameter of 
the cylinder was 10.5 inches and the stroke of the piston was 
12 inches. The clearance was ten per cent of the piston dis- 
placement. 

The indicator-diagram for a cut-off at one-fifth stroke, at 
which most of the investigations were made, is given by 
Fig. 75- From the indicated pressure the temperatures of 

* Proceedings of the Inst. Civ. Engrs., vol. cxxxil. 



344 



T HER MOD YNAMICS OF THE STEAM-ENGINE. 



the steam at various piston positions could be readily deter- 
mined by aid of a table of the properties of saturated steam. 





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55 5 10 15 20 25 30 35 

Time in sixtieths of a revolution from back end of stroke 



40 



45 



50 



Fig. 76. 



In Fig. 76 the indicated steam temperatures determined from 
Fig. 75 are plotted as ordinates, with sixtieths of a revolution 



INFLUENCE OF THE CYLINDER WALLS. 345 

for abscissae. The zero of abscissae is at the beginning of the 
forward stroke and the rise of temperature corresponds with 
the admission of steam shown on Fig. 75. The temperature 
of course remains constant for the part of the revolution which 
corresponds to the steam line of the indicator. There is then 
a gradual fall of temperature corresponding to the expansion, 
and a rapid drop corresponding to release. After release the 
back pressure temperature is constant till the beginning of 
compression, and during compression there is a rapid rise of 
temperature corresponding to the increase of pressure. The 
diagram Fig. 73 begins at ||- of the revolution in order that 
the most interesting events shall be recorded in the middle of 
the diagram. The crosses in the diagram show the tempera- 
ture of the steam as determined by the platinum thermometer 
in the piston of the engine. The divergency between the 
crosses and the full line representing indicated temperatures 
is considered by the investigators to be larger than the com- 
bined errors of the indicator and platinum thermometer. 
But at atmospheric pressure there are three degrees increase 
of temperature for each pound increase of pressure, and even 
at 70 pounds pressure there is one degree for a pound increase 
of pressure. Taking into consideration also the fact that the 
discrepancy is always on the side where the friction and lag 
of the indicator tend to place it, the discrepancy does not 
appear to require further explanation, unless it may be for 
the steam line; it is of course possible that there may be 
superheated steam in the middle of the cylinder during the 
admission of steam, together with a film of water on the cooler 
walls of the cylinder. 

The mean temperature of the cylinder-head is represented 
by a dotted line just above 300 F. The mean temperature 
of the steam is shown by a full line at 247 F. The dotted 
line lettered Metal cycle will be explained later. 

Fig. J J gives a similar diagram of indicated steam tempera- 
tures for another test, together with a diagram of temperatures 
by a platinum thermometer near the surface of the cylinder 



346 



THERMODYNAMICS OF THE STEAM-ENGINE. 



head. During all of the exhaust the steam near the cylinder 
head is strongly superheated, and compression carries the 
superheating far beyond the temperature of saturated steam 
during the admission. As steam comes into the cylinder the 
temperature near the cylinder-head falls rapidly till it passes 
below that of saturated steam just before cut-off. Though 
the temperature shown by the platinum thermometer con- 
tinues to fall after cut-off it soon rises beyond that shown by 



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Time in sixtieths of a revolution from back en'd of stroke 



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the indicator. This is considered by the investigators to 
show that the evaporation from the cylinder-head is probably 
complete soon after cut-off. Certainly the superheating dur- 
ing compression disposes of any assumption that there is any 
considerable amount of water in the cylinder at the beginning 
of compression. 

Fig. 78 gives a similar diagram of indicated steam tem- 
peratures, and also a diagram, represented by the dotted curve, 
of the changes of temperature near the inner surface of the 
cylinder. The scale for the full line representing indicated 
pressures is given at the left, the scale for the changes of the 
temperature of the wall is given at the right. The ther- 
moelectric couple by which the wall temperatures were deter- 



INFLUENCE OF THE CYLINDER WALLS. 



347 



mined was in the cylinder-head, and had a thickness of o.oi 
of an inch of imperforated wall between it and the inner 
surface. 

A comparison of the temperatures of the cylinder-head 
and of the side wall near the head is very instructive. In 
the first place the mean temperature of the side was I4°.3 
lower than that of the wall; and secondly, the temperature 
of the side varied more. Thus the range of variation for the 
head was 4°.9, while that for the side near the head was 
1 3°. 5. The general character of the cycle of temperature 























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Time in sixtieths of a revolution from back end of stroke 



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Fig. 78. 

changes for the side was similar to that for the head. The 
mean temperature of the cylinder-wall decreased from the 
head toward the crank end, as might be expected for a single- 
acting engine; and there was a considerable flow of heat in 
the same direction. A double-acting engine cylinder would 
differ in this respect, as the flow of heat would be from both 
ends to the middle, provided there were any perceptible flow. 
The temperature cycles for different points along the side of 
the cylinder were in general like the cycle for the end near 
the head; the greatest difference for a point beyond cut-off 
being due to its sudden exposure to steam when it was un- 



343 



THERMODYNAMICS OF THE STEAM-ENGINE. 



covered by the piston. Of course some difference in the 
cycle was produced by the fall of the mean wall temperature 
from the head toward the crank. 

The eight thermo-electric couples in the head, with thick- 
nesses of unbroken wall varying from o.oi of an inch to 0.64 
of an inch afforded means of determining the law of the flow 
of heat to and from the cylinder-wall. These and the pairs 
of couples along the side of the cylinder gave the means of 
calculating the heat absorbed by and restored by the cylinder. 
The methods of reducing the observations and making calcu- 
lations are too intricate to be given here; it will suffice to 
give some of the results. 

The following table gives the areas, temperatures, and the 
heat absorbed during a given test by the various surfaces 
exposed to steam at the end of the stroke, i.e., the clearance 
surface. 

Table VIII. 

CYCLICAL HEAT-ABSORPTION FOR CLEARANCE SURFACES. 



Portions of surface considered. 


Area 
of surface, 
square feet. 


Mean 

temperature, 

F. 


Heat absorbed 

B. T. U. 

per minute. 


Cover face, 10.5 inches diameter . . . 


0.60 
0.70 
O.OO 
O.I I 
O.71 
O.I2 
O.90 


305 

305 
295 
295 
297 
291 

305 


68 

79 
no 

20 


Piston face, 10.5 inches diameter . . 




123 

28 






102 








3-74 


30I 


530 





The heat absorbed by the side of the cylinder-wall uncov- 
ered by the piston up to 0.25 of the stroke was estimated to 
be 55 B. T. U. per minute, which added to the above sum 
gives 585 B. T. U. ; from which it appears that 90 per cent of 
the condensation is chargeable to the clearance surfaces. 
Further inspection shows that the condensation on the piston 
and the barrel is much more energetic than on the cover or 



INFLUENCE OF THE CYLINDER WALLS. 



349 



head. For example, the face of the piston absorbs no 
B. T. U. while the face of the cover absorbs only 68 B. T. u. ;. 
and the side of the cover and of the barrel, each 3 inches long, 
absorb 79 and 123 B. T. U., respectively. This relatively 
small action of the surfaces of the head indicates that less 
gain is to be anticipated from the application of a steam- 
jacket to the head than to the barrel of a steam-engine; tests 
on engines confirm this conclusion. 

The exposed surfaces at the side of the cylinder-head and 
the corresponding side of the barrel are due to the use of a 
deeply cored head which protrudes three inches into the 
counterbore of the cylinder, and which has the steam-tight 
joint at the flange of the head. It would appear from this 
that a notable reduction of condensation could be obtained 
by the simple expedient of making a thin cylinder-head. 

The final results of the investigations and the comparison 
of the condensation due to the heat absorbed by the walls of 
the cylinder are given in Table IX. Considering the intricacy 
and difficulty of the investigations the comparison of indicated 
and calculated condensations and evaporations must be con- 



Table IX. 

INFLUENCE OF THE WALLS OF THE CYLINDER. 
Callendar and Nicholson, Proc. Inst. Civ. Engrs., 1897. 



Duration, minutes 

Revolutions per minute 

Mean gauge-pressure 

Gross steam per revolution 

Leakage correction 

Net steam per revolution 

Steam caught at compression 

Weight of mixture in cylinder . . . 
Indicated steam at quarter stroke. 

Indicated steam at release 

Increase of indicated weight 

Adiabatic condensation 

Indicated evaporation 

Calculated evaporation 

Indicated condensation 

Calculated condensation 

Indicated horse-power 

Steam per H.P. per hour, pounds 



I. 


II. 


III. 


IV. 


V. 


VI. 


37 


68 


55 


79 


76 


35 


43.8 


45-7 


47-7 


70.4 


73-4 


81.7 


87.9 


89.2 


94.4 


98.1 


92.0 


94.2 


0.1422 


0-1437 


0.T483 


0.1094 


0. 1036 


0. 1000 


0.1004 


0.0976 


0.0990 


0.0697 


0.0627 


0.0576 


0.0418 


0.0461 


0.0493 


0.0397 


0.0409 


0.0424 


0.0107 


0.0104 


0.0103 


. 0099 


. 0098 


O.OIOO 


0.0525 


0.0565 


0.0596 


0.0496 


0.0507 


0.0524 


. 0407 


0.0414 


0.0437 


0.0418 


0.0394 


. 0408 


0.0466 


0.0456 


0.0488 


0.0460 


0.0436 


0.0454 


0.0059 


0.0042 


. 005 1 


. 0042 


0.0042 


. 0046 


0.0019 


0.0020 


0.0021 


0.0020 


0.0019 


0.0020 


0.0078 


0.0062 


0.0072 


0.0062 


0.0061 


0.0066 


0.0076 


0.0073 


0.0072 


. 0048 


0.0046 


0.0041 


0.0118 


0.0151 


0.0159 


0.0078 


0.0113 


0.0116 


0.0148 


0.0142 


0.0136 


0.0092 


. 0089 


0.0080 


4.10 


4-34 


4.78 


7.02 


6.67 


7.71 


26.8 


29.1 


ag-5 


23.8 


27.1 


26.9 



VII. 



25 
97.0 

96.0 
0.0856 
0.0494 
0.0362 
0.0105 
0.0467 

0.0393 

0.0426 

0.0033 

0.0019 
0.0052 

0.0035 
0.0074 
0.0067 

8.81 
23.8 



35° THERMODYNAMICS OF THE STEAM-ENGINE. 

ceded to be very satisfactory, and it must be admitted that 
the interchanges of heat are mainly, if not entirely, due to the 
metal of the cylinder-wall and not to water which remains in 
the cylinder from one stroke to the next. 

Leakage of Valves. — Preliminary tests when the engine 
was at rest showed that the valve and piston were tight. 
The valve was further tested by running it by an electric 
motor when the piston was blocked, the stroke of the valve 
being regulated so that it did not quite open the port, where- 
upon it appeared that there was a perceptible but not an im- 
portant leak past the valve into the cylinder. There was also 
found to be a small leakage past the piston from the head to 
the crank end. 

But the most unexpected result was the large amount of 
leakage past the valve from the steam-chest into the exhaust. 
This was determined by blocking up the ports with lead and 
running the valve in the normal manner by an electric motor. 
This leakage appeared to be proportional to the difference of 
pressure causing the leak, and to be independent of the num- 
ber of reciprocations of the valve per minute. From the tests 
thus made on the leakage to the exhaust, the leakage correc- 
tion in Table IX was estimated. Although the investigators 
concluded that their experimental rate of leakage was quite 
definite, it would appear that much of the discrepancy between 
the indicated and calculated condensation and vaporization 
can be attributed to this correction, which was two or three 
times as large as the weight of steam passing through the 
cylinder. Under the most favorable condition (for the 
seventh test) the leakage was 0.0494 of a pound per stroke, 
and since there were 97 strokes per minute, it amounted to 

0.0494 X 97 X 60 = 287.5 

pounds per hour, or 32.6 pounds per horse-power per hour, 
so that the steam supplied per horse-power per hour amounted 
to 56.4 pounds. If it be assumed that the horse-power is 
proportional to the number of revolutions, then the engine 



INFLUENCE OF THE CYLINDER WALLS. 35 1 

running double-acting will develop about 44 horse-power per 
hour, and the leakage then would be reduced to 6.5 pounds 
per horse-power per hour. Such a leakage would have the 
effect of increasing the steam-consumption from 23.5 to 30 
pounds of steam per horse-power per hour. 

To substantiate the conclusions just given concerning the 
leakage to the exhaust, the investigators made similar tests 
on the leakage of the valves of a quadruple-expansion engine, 
which had plain unbalanced slide-valves. The valves chosen 
were the largest and smallest; both were in good condition, 
the largest being absolutely tight when at rest. Allowing 
for the size and form of the valve and for the pressure, sub- 
stantially identical results were obtained. 

The following provisional equation is proposed for calcu- 
lating the leakage to the exhaust for slide-valves: 

leakage = — , (307) 

where / is the lap and e is the perimeter of the valve, both in 
inches, and/ is the pressure in pounds in the steam-chest in 
excess of the exhaust pressure. The valve of the constant in 
equation (307) is 0.021 for the high-speed engine used by 
Callendar and Nicolson, and is 0.019 f° r one test each of the 
valves for the quadruple engine, while another test on the 
large valve gave 0.021. 

This matter of the leakage to the exhaust is worthy of 
further investigation. Should it be found to apply in general 
to slide-valve and piston-valve engines it would go far towards 
explaining the superior economy of engines with separate 
admission- and exhaust-valves, and especially of engines with 
automatic drop-cut-off valves which are practically at rest 
when closed. It may be remarked that the excessive leakage 
for the engine tested appears to be due to the size and form 
of valves. The valve was large so as to give a good port- 
opening when the cut-off was shortened by the fly-wheel 
governor, and was faced off on both sides so that it could slide 



35 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

between the valve-seat and a massive cover-plate. The cover- 
plate was recessed opposite the steam-ports, and the valve was 
constructed so as to admit steam at both faces; from one the 
steam passed directly into the cylinder and from the other it 
passed into the cover-plate and thence into the steam-port. 
This type of valve has long been used on the Porter-Allen 
and the Straight-line engines; the former, however, has 
separate steam- and exhaust-valves. Such a valve has a very 
long perimeter which accounts for the very large effect of the 
leakage. 

Messrs. Callendar and Nicolson consider that the leakagj 
is probably in the form of water which is formed by conden- 
sation of steam on the surface of the valve-seat uncovered by 
the valve, and say further, that it is modified by the condi- 
tion of lubrication of the valve-seat as oil hinders the leakage. 






CHAPTER XV. 
ECONOMY OF STEAM-ENGINES. 

THE importance and the intricacy of the action of the 
walls of the cylinder of a steam-engine have thus far prevented 
any formulation of the actual economy of steam-engines. It 
therefore is necessary to make a study of steam-engine tests 
to learn the conditions favorable to economy and to determine 
£he effects of various devices and methods employed for the 
purpose of improving economy. 

Table X gives the economy of various types of engines, 
and represents the present state of the art of steam-engine 
construction. It must be considered that in general the 
various engines for which results are given in the table were 
carefully brought up to their best performance when these 
tests were made. In ordinary service these engines under 
favorable conditions may consume five or ten per cent more 
steam or heat; under unfavorable conditions the consumption 
may be half again or twice as much. 

All the examples in the table are taken from reliable 
tests: a few of these tests are stated at length in the chapter 
on the influence of the cylinder- walls; others are taken from 
various series of tests which will be quoted in connection with 
the discussion of the effects of such conditions as steam- 
jacketing and compounding; the remaining tests will be given 
here, together with some description of the engines on which 
the tests were made. 

The first engine named in the table is at the Chestnut Hill 
pumping-station for the city of Boston. Its performance is 

353 



354 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table X. 

EXAMPLES OF STEAM-ENGINE ECONOMY. 



Type of Engine. 


u 

<u 
a 

05 

a 
.2«j 

3 3 

o.S 

<u 3 


Steam-pressuie. 
pounds per 
square inch. 



0. 

<L> 
Ih 

O 

hi 


Steam per horse- 
power per hour. 
Pounds. 


4> -• 
<A G 

xi 
03 


Coal per horse- 
power per hour, 
pounds. 


Triple-expansion engines: 

Leavitt pumping - engine at Chestnut 
Hill 


50. 6 

56 

20.3 

384 

92 
61 
72 
94 

18.6 

13.2 

76 

89 

71 

56 

60 

59 
76 
5i 
60 

61 
61 
76 
126 

61 

*90 
*50 

*n 

*2.6 


176 
149 
122 
I70 

147 
165 
145 
154 

137 

99 

159 

92 

69 

57 

84 
61 
96 
72 
69 

104 
78 
96 

120 

77 

47 
59 


576 
T823 

574 
30 

125 

645 

1994 

1136 

643 

252 

595 

78 

266 

37i 

176 
150 

145 

180 

259 

178 

209 

120 

80 

16 

4i 
6.8 

8.8 
1.6 


II. 2 

n-3 
11. 8 
12.7 

13-7 
13-4 
150 

15-5 

12.2 

13-9 

13 

16.9 

18.4 
21.2 

16.9 
18. 1 
19.4 
20.5 
21.9 

22.1 
24.2 

23-9 
25.6 

33-5 

67 
125 

9* 

243 


204 
218 

231 

263 

222 

548 

IIIO 
2070 


I- 15 

1. 19 
I.25 


Allis pumping-engine at Milwaukee.... 


Experimental engine at the Massachu- 






I.46 
2 .OI 






2 


Compound engines : 

Leavitt pumping-engine at Louisville. . 
Leavitt pumping - engine at Boston 


i-33 








2.45 
2.66 




Simple engines, condensing : 








Harris-Corliss engine at Cincinnati. . . . 








Simple engines, non-condensing : 








Harris-Corliss engine at Cincinnati. . . . 
Hoadley portable engine 


3-35 


Harris-Corliss engine at the Massachu- 

Direct-acting steam-pumps : 

Fire-pump at the Massachusetts Insti- 

Steam- and feed-pump on the Minne- 


do. at reduced power 





* Strokes per minute. 



ECONOMY OF STEAM-ENGINES. 355 

the best known to the writer for engines using saturated 
steam. Some engines using superheated steam have a less 
steam-consumption than this engine, but their economy is not 
given here for reasons which will appear in the discussion of 
the use of superheated steam. This engine, which was 
designed by Mr. E. D. Leavitt, has three vertical cylinders 
with their pistons connected to cranks at 120 . Each cylinder 
has four gridiron valves, each valve being actuated by its own 
cam on a common cam-shaft; the cut-off for the high-pressure 
cylinder is controlled by a governor. Steam-jackets are 
applied to the heads and barrels of each cylinder, and tubular 
reheaters are placed between the cylinders. Steam at boiler- 
pressure is supplied to all the jackets and to the tubular 
reheaters. 

Table XI. 

TRIPLE-EXPANSION LEAVITT PUMPING-ENGINE AT THE 
CHESTNUT HILL STATION, BOSTON, MASSACHUSETTS. 

CYLINDER DIAMETERS 13-7, 24.375, AND 39 INCHES; STROKE 6 FEET. 

By Professor E. F. Miller, Technology Quarterly, vol. ix, p. 72. 

Duration, hours 24 

Total expansion 21 

Revolutions per minute 50.6 

Steam-pressure above atmosphere, pounds per square inch 175-7 

Barometer, pounds per square inch . 14.9 

Vacuum in condenser, inches of mercury 27.25 

Pressure in high and intermediate jacket and reheaters, pounds 

per square inch 175-7 

Pressure in low-pressure jacket, pounds per square inch 99.6 

Horse-power 575-7 

Steam per horse-power per minute, pounds 11. 2 

Thermal units per horse-power per minute 204.3 

Thermal efficiency of engine, per cent 20.8 

Efficiency for non-conducting engine, per cent 28.0 

Ratio of efficiencies, per cent 74 

Coal per horse-power per hour, pounds 1.146 

Duty per 1,000,000 b. t. u 141,855,000 

Efficiency of mechanism, per cent 89.5 

The Sulzer engine at Augsburg has four cylinders in all, 
a high-pressure, an intermediate, and two low-pressure cylin- 



35^ 



THERMODYNAMICS OF THE STEAM-ENGINE. 



ders. The high-pressure cylinder and one low-pressure 
cylinder are in line, with their pistons on one continuous rod, 
and the intermediate cylinder is arranged in a similar way 
with the other low-pressure cylinder. The engine has two 
cranks at right angles, between which is the fly-wheel, grooved 
for rope-driving. Each cylinder has four double-acting 
poppet-valves, actuated by eccentrics, links, and levers from 
a valve-shaft. The admission-valves are controlled by the 
governors. Four tests were made on this engine, as recorded 
in Table XII. 



Table XII. 

TRIPLE-EXPANSION HORIZONTAL MILL-ENGINE. 

CYLINDER DIAMETERS 29.9, 44.5, AND TWO OF 51.6 INCHES; STROKE 78.7 

INCHES. 

Built by Sulzer of Winterthur, Zeitschrift desVereins Deutscher Ingenieure y 

vol. xl, p. 534- 



IV 



Duration, minutes 

Revolutions per minute 

Steam-pressure, pounds per square inch 

Vacuum, inches of mercury 

Horse-power 

Steam per horse-power per hour, pounds 

Mean for four tests 11.46 

Coal per horse-power her hour, pounds. 

Mean for four tests 1.30 

Steam per pound of coal 



I 


II 


III 


306 


322 


272 


56.23 


56.28 


56.18 


145.4 


147.9 


148.4 


27.24 


27.20 


27.20 


1872 


1835 


1850 


n-53 


11.49 


11.49 


i-37 


1.36 


1.29 


8.78 


8.49 


8.97 



327 
56.18 

149-0 
27.19 
1823 
n-33 

1. 19 
9.62 



In this connection there is given in Table XIII the details 
of five tests made on an engine by the same builders, but of 
smaller size and having but three cylinders. The high-pres- 
sure and intermediate pistons are on one continuous piston- 
rod and the low-pressure piston is on a separate rod. The 
engine has two cranks at right angles and a fly-wheel between 
them. 

The details of the test on the Milwaukee pumping-engine 
are given in Table VI on page 332. The engine has three 



ECONOMY OF STEAM-ENGINES. 



357 



vertical cylinders with their pistons connected to cranks at 
120°. The valve-gear is of the Corliss type, controlled by a 
governor. 

Table XIII. 

TRIPLE-EXPANSION HORIZONTAL MILL-ENGINE AT 

AUGSBURG. 

CYLINDER DIAMETERS 11.28, 17-75, AND 27. 6l INCHES; STROKE 39.37 INCHES. 

By Professor M. Schroter, Zeitschrift des Vereins Deutscher Ingenieure, 

vol. xxxiv, p. 7. 



V 



Duration, minutes 

Revolutions per minute 

Cut-off high-pressure cylinder 

Total expansion 

Steam-pressure, pounds per square inch 
above atmosphere 

Barometer, pounds per square inch 

Back - pressure, absolute, pounds per 
square inch 

Condensation in jackets in per cent of 
total steam-consumption 

Condensation withdrawn from first in- 
termediate receiver, per cent 

Horse-power 

Steam per horse-power per hour, pounds 

B. T. U. per horse-power per minute. . . . 



I 


II 


III 


IV 


324 


306 


330 


301 


70.5 


70.2 


70.3 


76.3 


0.259 


0.252 


0.259 


0.308 


25.6 


26.3 


25-6 


21.5 


145.6 


146.4 


147.6 


150.6 


13*9 


13-9 


13-8 


13-8 


I.I 


1.2 


1-3 


1.2 


16. 1 


20.0 


18.3 


17.O 


2.7 


2.9 


2.7 


3-o 


195.0 


198. 1 


198. 1 


222.5 


12.61 


12.20 


12.92 


12.72 


226 


221 


230 


227 



326 
70.4 

0.300 

22.1 
143.2 

13-9 
i-3 

18. 1 

2.2 
212.8 

12.94 
230 



The test on the Willans engine is taken from Table XLII 
on page 406. This remarkable engine, though of small size 
and power, gives an exceedingly good economy and is adapted 
to driving high-speed machinery directly. The three pistons 
are arranged on one rod and are single-acting. The under 
sides of the high-pressure and the intermediate pistons form 
the tops of the intermediate receiver-spaces, so that in a 
manner the expansion has five stages. 

The test on the experimental engine at the Massachusetts 
Institute of Technology is quoted here because its efficiency 
and economy are chosen for discussion in Chapter XL 
Taking its performance as a basis, it appears on page 248 that 



35^ THERMODYNAMICS OF THE STEAM-ENGINE. 

with 150 pounds boiler-pressure and 1.5 pounds absolute 
back-pressure an engine may be expected to give a horse- 
power for 1 1.5 pounds of steam, from which it appears that 
under its conditions its performance compares favorably with 
the Sulzer engine or even the Leavitt engine. 



Table XIV. 

MARINE-ENGINE TRIALS. 

By Professor Alexander B. W. Kennedy, Proc. Inst. Mech. Engs., 1889- 
1892 ; summary by Professor H. T. Beare, 1894, p. 33. 



Triple or compound 

Diameter high-pressure cylinder, inches 

Diameter intermediate cylinder, inches 

Diameter low-pressure cylinder, inches 

Stroke, inches .... 

Duration of trial, hours 

Number of expansions 

Revolutions per minute 

Steam-pressure above atmosphere, pounds per 

square inch 

Pressure in condenser, absolute, pounds per 

sqare inch 

Back-pressure, absolute, pounds per sq. in. . . . 

Horse-power 

Steam per horse-power per hour, pounds 

Thermal units per horse-power per minute 

Coal per horse-pcwer per hour, pounds 

Steam evaporated per pound of coal 

Weight of machinery per horse-power, pounds. 



C 

27.4 



50.3 
33 
14 
6.1 

55-6 

56.8 

2.32 

3-8 

37i 
21 .2 
380 
2.66 
7.96 
603 



C. 

3o 



> 
•a o 



a 

50.1 



57 

36 

10.9 

6.1 

86 

80.5 

2.51 
3-4 
1022 
21.7 

393 
2.9 

7-4Q 

448 



97.1 
72 

9 

5-7 
36 

105.8 

4.72 
6.0 

2977 

20.8 

367 
2-3 
8.97 

272 



T. 

29.4 

44 
70. 1 

48 

17 
10.6 

71.8 

145.2 

2-73 

3-3 
1994 

15.0 
265 

2.01 

7.46 
439 



T. 

21.9 

34 

57 

39 

16 
19.0 
61. 1 

165 

0.70 

1.8 
645 

13-4 
250 

1.46 

9-15 
701 



The engines of the S. S. Iona have an unusually large 
expansion and give a correspondingly good economy. The 
engines of the Meteor and of the Brookline give the usual 
economy to be expected from medium-sized marine engines. 
Table XIII gives details of tests on the engines of the two 
first ships mentioned, together with tests on compound 
marine engines. Table XV gives tests on the engine of the 
Brookline. It appears probable that the relatively poor 



ECONOMY OF STEAM-ENGINES. 



359 



economy of marine engines compared with stationary engines 
is due to the smaller degree of expansion, which is accepted 
to avoid using large and heavy engines. 

Table XV. 

TESTS ON THE ENGINE OF THE S. S. BROOKLINE. 

CYLINDER DIAMETERS 23, 35, AND 57 INCHES | STROKE 36 INCHES. 

By F. T. Miller and R. G. B. Sheridan, Thesis, 1895, M.I.T. 



Duration, hours 

Revolutions per minute 

Steam-pressure, pounds per square inch above 

atmosphere 

Vacuum, inches of mercury 

Horse-power 

Steam per horse-power per hour, pounds 

Coal per horse-power per hour, pounds 

B. T. U. per horse-power per minute 



I 


II 


III 


IV 


2 


2 


1 


3i 


94.6 


93-6 


93-6 


93 


155 


155 


154 


145 


21.6 


21.0 


22.2 


21 .7 


1242 


1221 


1136 


ii37 


17.2 


16.9 


15-5 


17.0 


2.22 


2.17 


1.99 


2.18 


292 


288 


263 


288 



2£ 

93 



148 
20.9 

1 148 
16.3 

2.09 

277 



Table XVI. 

COMPOUND LEAVITT PUMPING-ENGINE AT LOUISVILLE, 

KENTUCKY. 

CYLINDER DIAMETERS 27-2 AND 54. 1 INCHES ; STROKE 10 FEET. 

By F. W. Dean, Trans. Am. Soc. Mech. Engs., vol. xvi, p. 169. 

Duration, hours 144 

Revolutions per minute 18.6 

Pressure, pounds per square inch : 

Barometric 14.6 

Boiler above atmosphere 140 

At engine above atmosphere 137 

Back-pressure, 1. p. cylinder 0.95 

Total expansions « < 20 

Moisture in steam, per cent 0.55 

Horse-power 643.4 

Steam per horse-power per hour, pounds 12.2 

B. T. U. per horse-power per minute 222 

Thermodynamic efficiency, per cent 19 

Mechanical efficiency, per cent 93 

The best performance of a compound engine using satu- 
rated steam, which is known to the writer, is that reported 
for a pumping-engine at Louisville, Ky. This engine has 



360 



THE R MOD YNAMICS OF THE STEAM-ENGINE. 



two cylinders, each jacketed with steam at boiler-pressure on 
barrels and heads and steam at the same pressure is used in a 
tubular reheater. Each cylinder has four gridiron valves 
actuated by as many cams on a cam-shaft. The details of 
the test are given in Table XVI, and Table XVII gives two 
tests on another compound Leavitt engine at the Boston 
main-drainage pumping-station. A comparison of the two 
tables shows the advance made in ten years from 1885 to 
:8 95 . 

Table XVII. 

LEAVITT COMPOUND PUMPING ENGINE AT THE BOSTON 
MAIN-DRAINAGE WORKS. 

CYLINDER DIAMETERS 25.5 AND 52 INCHES J STROKE 9 FEET. 



Duration, hours and minutes 

Revolutions per minute 

Boiler-pressure above atmosphere, pounds per square inch. . 

Vacuum, inches of mercury . . . , 

Barometer, inches of mercury 

Horse-power 

Steam per horse-power per hour, pounds 

Coal per horse-power per hour, pounds 

Steam per pound of coal, from and at 212 F 

Duty, work per 100 pounds of coal, millions of foot-pounds... 



24-43 


24-31 


13.17 


13.42 


994 


98.6 


28.1 


28.0 


30.18 


29.81 


251.5 


290.2 


13-9 


14.2 


i-33 


i-35 


12.12 


11.83 


122.5 


122.4 



Table XVIII. 

CROSS-COMPOUND MILL-ENGINE AT NATICK, R. I. 

CYLINDER DIAMETERS 18.4 AND 48.5 INCHES J STROKE 4 FEFT. 

By F. W. Dean, Trans. Am. Soc. Mech. Engs,, vol. xvi, p. 179. 



Duration, hours 

Revolutions per minute 

Steam-pressure above atmosphere, pounds per square inch. 

Vacuum, inches of mercury 

Total expansions 

Moisture in steam , per cent 

Horse-power 

Steam per horse-power per hour, pounds 



II 



4-5 


5 


76.4 


76.6 


159 


158 


25-4 


25-2 


33 


33-4 


1.9 


1.8 


595 


582 


13 


13.2 



The compound mill-engine at Natick, R. I., has two 
cylinders with their pistons acting on cranks at right angles. 



ECONOMY OF STEAM-ENGINES. 36 1 

The valves are of the gridiron type as made by the Wheelock 
Engine Company. The high-pressure cylinder is jacketed on 
the heads and barrel; the low-pressure cylinder is jacketed 
on the heads only. A reheater is placed between the cylin- 
ders. A notable feature of this engine is the ratio of the 
volume of the cylinder, 



48.5 : 18.4 : : ; : I, 

which gives a large number for the total expansion without 
requiring a very early cut-off for the high-pressure cylinder. 

Table XIX gives tests on another cross-compound engine 
which has a cylinder ratio of three and a half to one, and 
which has neither steam-jackets nor reheater, and which has 
nevertheless nearly as good an economy. 

Table XIX. 

CROSS-COMPOUND MILL-ENGINE AT NEW BEDFORD, MASS. 

CYLINDER DIAMETERS 30 AND 56 INCHES ; STROKE 72 INCHES. 

By Denton, Jacobus, and Rice, Trans. Am. Soc. Mech. Engs., vol. xv, 

p. 882. 

Revolutions per minute 65.2 

Steam-pressure above atmosphere, pounds per square inch 123.0 

Vacuum, inches of mercury 25.6 

Total expansions 13.4 

Superheating at throttle, degrees F. 14.6 

Horse-power 1592 

Steam per horse-power per hour, pounds 13.5 

Coal per horse-power per hour, pounds l^- 

B. T. U. per horse-power per minute 247 

Table XX gives tests on an engine built by the same 
makers as that for which tests are recorded in Table XVIII. 
This engine has three cylinders with a ratio of seven to one 
for the volumes of the largest and smallest cylinders. In two 
of the tests the engine was run as a compound engine using 
only the smallest and largest cylinders; in the other two tests 
the three cylinders were used as designed. The engine has 
jackets on the heads and barrels of the high-pressure and 



362 



THERMODYNAMICS OF THE STEAM-ENGINE. 



intermediate cylinders and on the heads of the low-pressure 
cylinder. A further consideration of these tests will come in 
connection with the discussion of compound and triple- 
expansion engines. 

Table XX. 

HORIZONTAL MILL-ENGINE AT HOLYOKE, MASS. 

DIMENSIONS: HIGH-PRESSURE CYLINDER, 12 IN. DIAMETER, 36 IN. STROKE; 
INTERMEDIATE CYLINDER, l6 IN. DIAMETER, 36 IN. STROKE; 
LOW-PRESSURE CYLINDER, 1^\\ IN. DIAMETER, 48 IN. STROKE. 

Tested by S. M. Green and G. I. Rockwood, Trans. Am. Soc. Meek. Engs. t 

vol. xiii, p. 647. 

The engine was run with large and small cylinders only, and with all 
three. 



Duration, hours 

Revolutions per minute 

Steam-pressure above atmosphere, pounds 

per square inch 

Per cent of steam used in jackets. ....... 

Horse-power 

Steam per horse-power per hour, pounds. 



II 



Compound. 



5 
79.2 

142 

13 
187. 1 

13.4 



5 
79-3 

142 

14 

180.7 

I3-I 



III 



IV 



Triple. 



5 
79.0 

142 

16 
199. 1 

13-0 



5 
79.0 

143 
16 

178.2 
13-3 



Table XXI gives four tests on a small portable engine 
which gives a very good economy for its type and size. 

The details of the tests on the Rush and the Fust Yama 
are given in Table XXXII and XIV on pages 256 and 358. 

A remarkably complete and important series of tests was 
made in 1884 by M. F. Delafond. These tests are recorded 
in Tables XXX and XXXI, from which there are quoted in 
Table X four results with and without condensation and with 
and without steam in the jackets. 

The details of the tests on the Harris-Corliss engine at 
Cincinnati, together with tests on two similar engines, are 
given in Table XXII. 



ECONOMY OF sTEAto-ENGINES. 



363 



Table XXI. 

PORTABLE COMPOUND CONDENSING-ENGINE. 

Tested by a Committee of the Soc. Ind. de Mulhouse, 1879. 
Reported by Isherwood, Jour. Franklin Inst., vol. cxx. 



Duration, hours 

Revolutions per minute 

Cut-off, small cylinder 

Total expansions 

Steam-pressure above the atmosphere, 

lbs. per sq. in 

Atmospheric pressure, lbs. per sq. in... 

Back-pressure, lbs. per sq. in 

Horse-power, indicated 

Horse-power by brake 

Steam per I.H.P. per hour lbs 

B. T. U. per horse-power per hour 



3 
89 

0.42 
6.26 

91.8 
14.2 
2.4 
77-2 
67.7 
17.7 
3i8 



II 



4 

88.5 
0.42 
6.26 

9i-5 
14.2 
2.4 
77.6 
67-5 
17-3 
323 



III 



3 
90 
0.25 
9.64 

91. 7 

14.2 
2.4 
63-9 
55-7 
16.9 
318 



IV 



3-24 
88.7 
0.42 
6.26 

92.2 

14.2 

2.4 

77-8 

67-5 
16.9 

315 



Table XXII. 

AUTOMATIC CUT-OFF ENGINES. 

cylinder diameters 18 inches; stroke 4 feet. 

By J. W. Hill. 

(First Millers' International Exhibition, Cincinnati, 1880.) 



Duration 

Cut-off 

Revolutions per minute 

Boiler-pressure above atmos., lbs. per sq. in. 

Barometer, inches of mercury 

Vacuum, inches of mercury 

Back-pressure, absolute, lbs. per sq. in.... 

H orse-power 

Steam per horse-power per hour, pounds. . 
B. T. U. per horse-power per hour 



Condensin 


gr- 


Non 


-condensing. 


R. 


H. 


w. 


R. 


H. 


W. 


10 


10 


10 


9 


10 


10 


0. 124 


0.119 


0.131 


0.160 


0.136 


0.170 


75-4 


75-8 


74-5 


75-3 


75-8 


76.1 


95-8 


96.1 


9 6 -3 


96.6 


9 6 -3 


96.3 


2^.7 


29.6 


29.4 


29.8 


29.6 


29-5 


2 5-5 
4-5 


25.7 
3-4 


24.0 
4-7 








15-5 


14.9 


15-5 


143-2 


145- 1 


J 43-9 


121 .7 


119. 7 


126.7 


20.6 


19.4 


19-5 


25-9 


23-9 


24.9 


372 


349 


343 


433 


400 


4i5 



The tests on the engine of the Gallatin are taken from 
Table XXXII on page 388. 

Table XXlla gives the details of a test on a Hoadley 
portable engine which had a piston-valve controlled by a fly- 
wheel governor, made by Mr. Hoadley in 1876. 



3^4 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table XXlIa. 

HOADLEY PORTABLE ENGINE. 

CYLINDER DIAMETER I4.56 INCHES; STROKE 28.5 INCHES. 

Duration of test 6 hrs. 2 min. 

Revolutions per minute 125.96 

Steam-pressure in boiler, pounds 120 

Cut-off, per cent 17 

Horse-power by indicator 80.29 

Horse-power by brake 72.72 

Friction of engine, horse-power 7.57 

Horse-power by indicator without load 5.80 

Steam per I.H.P. per hour, weighed, pounds 25.61 

" brake H.P. per hour, weighed, pounds... 28.27 

Coal per indicated horse-power per hour, pounds... 3.35 

" " brake horse-power per hour, pounds 3.69 



Table XXIII. 

DUPLEX DIRECT-ACTING FIRE-PUMP AT THE MASSACHU- 
SETTS INSTITUTE OF TECHNOLOGY. 

TWO STEAM-CYLINDERS l6 INCHES DIAMETER, 12 INCHES STROKE. 







Techno 


logy Quarterly, vol. viii, p. 19 






<u 
a 

<u 
o 

.s's 

c/5 


u 



u 
v> 

O 

■Zi 

►J 



u 

Cfl 

S-) 


■5 

C nj 
4) M 


4> 

u 

p 

« 

u 

cx<u 

C/3 


09 

u 

U 

"O 

u.S 

Cfl 4> 
OCT) 


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Ih 

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v.' e 

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■ CD 

u: as 

1^ 

HI 


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°t! 

»- a 
S£ 

cS O 
<u 0. 


1 

u 

u a 

4> CU 

a-a 
Ed d 

hI 

CQ 


Duty. (Foot- 
pounds per 
1 ,000,000 B. T. U.) 


99 
114 
119 

135 
156 
193 
175 
180 


11.40 

IT . 70 
II.49 
II.60 
IO.9O 
IO.O9 
11-77 

H-74 


IO.IO 
II.O7 
II.O7 
II. IO 
I0.26 
IO.3I 
11.79 

11.66 


58 

55 
5i 
53 
47 
45 
45 
46 


5 
6 

4 
8 
2 
6 
6 
5 


6.78 

12.48 

12.18 
18.24 

21.00 
32-95 

39-55 
41.20 


I9.8O 


125 

IOI 

IO9 

92 

98 
78 

66 

67 


2070 
1674 
I809 
1530 
1619 
I29I 
IO83 
IIIO 


13,920,000 
17,540,000 
16,980,000 
19,850,000 
18,280,000 
23,730,000 
27,980,000 
27,030,000 



The test of the small Harris-Corliss engine at the Massa- 
chusetts Institute of Technology is taken from Table II on 
page 318. A complete calculation for the application of 
Hirn's analysis to this test is given on page 313. 



ECONOMY OF STEAM-ENGINES. 



365 



The two tests on the direct-acting fire-pump at the 
Massachusetts Institute of Technology are taken from Table 
XXIII, and the tests on the feed- and fire-pump on the 
Minneapolis are given in Table XXIV. Both sets of tests 
show the extravagant consumption of steam by such pumps 
when running at reduced powers. The latter table is most 
interesting on account of the light that it throws on the way 
that coal is consumed by a war-vessel when cruising at slow 
speeds or lying in harbor. 

Table XXIV. 

TESTS OF AUXILIARY STEAM MACHINERY OF THE U. S. S. 

MINNEAPOLIS. 

By P. A. Engineer W. W. White, U. S. N., Journal Am. Soc. Naval 

Enes., vol. x. 



Engine or pump tested. 



s 




rt 




4J 




*J 




X 


, 




Efl 





V 




X! 


■a 

c 


s 


>^ 


3 





£ 





8^ 

2 o 



Centre circulating-pump: 

Full power 2 

Reduced power* 2 

Starboard circulating-pump: 

Reduced power 2 

Starboard air-pump. ... 2 

Centre air-pumpt 2 

Water-service pump 2 

Fire- and feed -pump 2 

do. 2 

do. 2 

do. 2 

Fire- and bilge-pump 2 

Blower-engine 2 

Dynamo-engine 2 

do. 2 

Ice-machine engine 1 



16 
16 

7- 



12 
12 

5 
10.5 
10.5 

7 



to 












<u 






en a 






d-f 1 


V 






<u 


a = 




u 




5 


u 





^ oT 





O 


X3 


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. C 

2 


re M 

.Sx: 


rex: 

■— 


ouble s 
revolut 
minute 


a 



re 

V- 

3 


•a • 

II 

■a 0. 


Q 


2: 


<! 


Q 


Q 


~ 


36 


6 


6 


171. 6 


3-7 


18.9 


36 


6 


6 


90 


2-50 


4.1 


36 


6 


6 


82 


3-28 


2.0 


3i-5 


21 




16.6 


2- S 8 


6.5 


3i-5 


21 




15.2 


3-2 


25.2 


4-5 


10 


7-5 


40.9 


2-59 


1.04 


8-5 


12 


10.9 


12.7 


3-31 


0.78 


8.5 


12 


12.0 


37-3 


1-46 


6.4 


7-5 


12 


10 


Ii .O 


2-23 


8.8 


7-5 


12 


10.8 


2.6 


3-27 


1.6 


9 


12 


11. 2 


27.7 


2-2 


2.5 


.... 


4 


4 


595 


I-24 


16.3 




5 


5 


425 


I-IO 


22.9 




5 


5 


425 


0-26 


35-2 




10 


10 


73 - 1 


5-12 


6.0 



u 



76 



183 
78 
205 
319 
156 
91 
243 
171 

77 
65 
56 



* One cylinder only supplied with steam. 

+ Pump loaded with three times the power developed during official trial, when main 
engine indicated 7219 H. P. 

Effect of Raising Steam-pressure. — A study of the 
examples of steam-engine economy given in Table X, and of 
the details of the tests given in the several tables referred to, 
shows that in general a gain in economy is to be obtained by 
increasing the steam-pressure and the total number of expan- 



366 



THERMODYNAMICS OF THE STEAM-ENGINE. 



sions. This general statement can be held to be true only 
when the proper methods are used to ameliorate the effects of 
cylinder-condensation and reevaporation, which have already 
been seen to place a very strict limit on the number of 
expansions to be profitably used for a simple steam-engine. 

It is interesting here to investigate the effect of increasing 
the steam-pressure only, for a non-conducting engine, with- 
out changing the expansion. The accompanying table gives 
the efficiencies for such an engine for various steam-pressures 
calculated by equation (260), page 248, assuming in all cases 
that the absolute pressure at release is one-third that during 
admission. 

Table XXV. 



Pressure, pounds per square inch: 

By gauge during admission 

Absolute during admission 

At end of expansion 

Back-pressure 

Quality of steam at release 

Efficiency 

B. T. U. per horse-power per minute. ... 
Steam per horse-power per hour, pounds 



I 


II 


III 


30-3 


60.3 


90-3 


45 


75 


105 


15 


25 


35 


14.7 


14.7 


14.7 


0.948 


0.936 


0-934 


0.085 


O.III 


0.123 


499 


382 


345 


30.4 


22.6 


20.7 



IV 



120.3 
135 

45 
14.7 

o.933 
0.133 
319 
19.0 



From this calculation it appears that there is at first a 
notable gain in economy from increasing the steam-pressure 
without changing the expansion for a non-conducting engine, 
but that after a moderate pressure has been reached the 
gain for further increase is small. Now the actual engine 
will have a greater steam-consumption on account of the loss 
by external radiation, and the loss due to initial condensa- 
tion, reevaporation, and exhaust-waste. The absolute loss 
from external radiation increases with the steam-pressure, 
but the horse-power of the engine increases more rapidly; 
consequently the radiation per horse-power decreases as the 
pressure is increased. Since the changes of temperature are 
larger for a high than for a low pressure, it is a reason- 



ECONOMY OF STEAM-ENGINES. 367 

able inference that the condensation and reevaporation in 
the cylinder, and the exhaust-waste are liable to increase 
as the steam-pressure is raised. We may, therefore, expect 
to find that tests on steam-engines will give a result similar 
to that calculated for a non-conducting engine, i.e., that 
with fixed expansion there will be at first a considerable 
gain from raising the steam-pressure in a simple engine, 
but that beyond a moderate steam-pressure the gain will 
be very slow. Some tests even show a loss from raising the 
steam-pressure too high. 

A confirmation of the ideas just enunciated may be found 
in Table XXXVIII, page 403, which gives the results of 
tests on the Wiilans engine, using one cylinder only as a 
single-acting simple engine. Tests 1 to 6 were made with the 
cut-off at half stroke and with the steam-pressure varying from 
64.7 to 20 pounds absolute, the back-pressure being in all 
about one pound. Increasing the pressure from 20 to 55 
pounds absolute, or from about 5 to about 40 pounds above 
the atmosphere, gave a gain of about ten per cent, but a 
further increase to 64.7 pounds absolute, or about 50 pounds 
by the gauge, gave a distinct loss. A similar though not so 
pronounced effect is shown by tests 7 to 12. 

A more striking confirmation is given by tests on a Corliss 
engine at Creusot without steam in the jackets, as represented 
by Tables XXX and XXXI on pages 454 and 455 and by 
Fig. 79. In the figure the curves lettered C represent tests 
made with a vacuum and curves lettered N represent tests 
without a vacuum. The figures used in conjunction with the 
curves give the approximate boiler-pressure for tests repre- 
sented. The curves representing tests with condensation 
show a decided gain when the steam-pressure is raised from 
35 pounds to 50 pounds above the atmosphere, and an 
appreciable gain for a further increase from 50 to 60 pounds. 
A further increase to 80 and 100 pounds is accompanied 
by a distinct loss. The tests without condensation show a 
large gain from raising the pressure from 50 to 75 pounds 



368 THERMODYNAMICS OF THE STEAM-ENGINE. 

and little if any gain from a further increase to ioo pounds 
above the atmosphere. 

Methods of Improving Economy. — Bearing in mind the 
fact just established that there is little if any gain in economy 
to be obtained by raising the steam-pressure beyond a certain 
very moderate limit unless the expansion may be increased 
at the same time, and also the fact that the economical expan- 
sion for a simple unjacketed steam-engine is very strictly 
limited by initial condensation, reevaporation, and exhaust- 
waste, we are led to the consideration of methods for 
ameliorating the action of the walls of the cylinder. Two of 
these have been considered in connection with the study of 
the influence of the cylinder-walls in the preceding chapter, 
namely, the use of superheated steam and the use of the 
steam-jacket. Another method of even more practical im- 
portance is the use of compound and multiple-expansion 
engines. Again, reheaters may be placed between the cylin- 
ders of compound or multiple-expansion engines to dry or 
superheat the steam on its way from one cylinder to another. 
Any two or all of these several methods may be used in con- 
junction. Examples of almost all of the possible combinations 
may be found in practice, and the advantages to be obtained 
from some of them can be determined from tests which will 
now be studied in detail. 

Superheated Steam. — The most direct and effective way 
of improving steam-engine economy is by the use of super- 
heated steam, and yet no permanently good results have been 
obtained, on account of the practical difficulties of maintaining 
the superheating apparatus. Some upright boilers give 
superheated steam, but when the superheating is forced to 
any very considerable degree they are likely to be troubled by 
wasting of the upper ends of the tubes. Whenever super- 
heated steam has been used so as to give a notable gain in 
economy the superheating has been accomplished in a separate 
apparatus, which has taken the form of a coil of pipe exposed 
to the products of combustion beyond the boiler. Now it is 



ECONOMY OF STEAM-ENGINES. 369 

the accepted experience of boiler-makers that surfaces exposed 
to the fire must be of moderate thickness or they will rapidly 
waste away. Thus it is not desirable to make furnace-flues 
more than half an inch thick, and if they are made thicker 
they are liable to waste away till they are reduced to about 
that thickness. Plates and tubes if thin enough endure long 
service in a boiler when exposed to the fire because they are 
kept at a moderate temperature by the water in the boiler. 
If steam is to be superheated strongly in a coil of pipe or 
other device which is exposed to hot gases, the metal of the 
superheater must be strongly heated and is sure to waste 
away rapidly. There is no material that can stand long 
service when exposed at once to a high pressure and a high 
temperature. There is little risk, therefore, in predicting that 
all superheating devices now used will eventually be discarded 
for this reason. 

From the considerations just stated no tests with super- 
heated steam are given in Table X, though the economy 
obtained by the use of superheated steam is remarkable, both 
comparatively and absolutely. Several series of tests on 
engines using superheated steam will be given, together with 
a discussion of the advantages obtained from its use. 

Tests on the Eutaw. — The U. S. S. Entaw was built for 
special service in 1863-64, and had a single-cylinder inclined 
engine with poppet-valves and a Stevens adjustable cut-off. 
The steam from the boilers could be supplied directly to the 
engine or it could be passed through a coil superheater in the 
uptake. The tests were made at the dock, the speed of rotation 
being controlled by removing more or less of the paddles from 
the paddle-wheels. The details of the tests are given in 
Table XXVI. Four of the tests are with saturated steam 
and five are with superheated steam. The best result with 
saturated steam is obtained for a cut-off at 0.32 of the stroke, 
agreeing with the conclusions from tests on the MicJiigan 
(Table I, page 303). The tests with superheated steam show 
a good deal of irregularity, which cannot be satisfactorily 



37Q 



THERMODYNAMICS OF THE STEAM-ENGINE. 



accounted for either by the degree of superheating or the 
perfection of the vacuum, A comparison of the best result 
for superheated steam with the best result for saturated steam 
shows a gain in steam economy of 



30.6— 25.1 
30.6 



X 1 00 = 18 per cent, 



and a gain in coal-consumption of 
2.84 — 2.42 



2.84 



X 100 =15 per cent. 



Table XXVI. 

TESTS ON THE ENGINE OF THE U. S. S. EUTAW. 

CYLINDER DIAMETER 4 FEET 10 INCHES; STROKE 8 FEET 9 INCHES. 

By Chief Engineer Isherwood, Researches in Experimental Steam 

Engineering. 



Duration, hours 

Cut-off 

Revolutions per minute 

Initial pressure in the cylinder per 

square inch, absolute 

Barometer, inches of mercury 

Vacuum, inches of mercury 

Back-pressure, pounds per square 

inch, absolute 

Temp, of superheated steam 

Horse-power 

Steam per horse-power per hour, 

pounds 

Combustible per horse-power per 

hour, pounds 

Per cent of water in cylinder at 

release 



Saturated steam. 




Superheated 


steam. 


I 


II 


III 


IV 


V 


VI 


VII 


VIII 


72 
0.24 

5-49 


72 
0.32 

6.58 


72 

0.50 
8.60 


72 

0.58 
9.19 


72 

0.29 
6.48 


72 
0.32 

6-55 


72 

0.50 
9.00 


72 
0.50 

9-*5 


40.3 

3° 

27 


40.6 

30 
27.6 


40.9 

30 
27.1 


38.2 
29.7 
28 


41.0 
30.1 
28 


41.1 

30 

26.2 


41.7 
30 

26.2 


41 .0 

30. 1 
28.5 


1.8 
154-4 


i-3 

228.6 


1.6 
37 2 -5 


0.9 
414.9 


1.1 
39 6 
207.3 


2-5 

366 
218.2 


1.9 

358 

391.0 


0.9 

394 
406.8 


39-6 


30.6 


32-7 


3 x -4 


30.1 


29.2 


27.8 


25.1 


3-77 


2.84 


2.87 


2.90 


2.84 


2.64 


2.55 


2.42 


56 


37 


30 


24 


43 


37 


18 


21 



72 

0.58 
9.46 

41.2 

29.9 
28.3 

0.8 
392 

453-2 
27.1 

2.72 
17 



There is a very considerable reduction of the water in the 
cylinder at release, which explains sufficiently the reason for 
this notable gain in economy. 

Dixwell's Tests. — A small Harris-Corliss engine was 
fitted up for making tests on superheated steam at the 
Massachusetts Institute of Technology by Mr. George B. 
Dixwell. Six tests with superheated and saturated steam 
were made on this engine in 1877 in the presence of a board 



ECONOMY OF STEAM-ENGINES. 



371 



of United States naval engineers. The steam was generated 
in cylindrical tubular boilers and was superheated in a vertical 
boiler with a detached brick furnace. Three different points 
of cut-off were tried for each condition of steam; the cut-off 
was controlled by the governor in the usual way when the 
cut-off was less than half stroke; when it was more than half 
stroke the valve was not released by the drop cut-off 
mechanism and the cut-off was produced by the lap of the 
valve as for a plain slide-valve. 

Table XXVII. 

DIXWELL'S TESTS ON SUPERHEATED STEAM. 

CYLINDER DIAMETER 8 INCHES; STROKE 2 FEET. 

Proceedings of the Society of Arts ■, Mass. Inst. Tech., 1887-88. 



Duration, minutes 

Cut-off 

Revolutions per minute 

Boiler-pressure above atmosphere, pounds 

per square inch 

Back-pressure, absolute, pounds per sq in. 
Temperatures Fahrenheit: 

Near engine 

In cylinder by pyrometer 

Per cent of water in cylinder: 

At cut-off 

At end of stroke 

Horse-power 

Steam per horse-power per hour, pounds.. 
B. T. U. per horse-power per minute 



Saturated steam. 



127 
0.217 
61.5 

So-4 
15-4 

302 
278-297 

52.2 
32-4 

7-65 
48.2 
796 



II 



83 

0.443 
60.4 



50.2 
15-7 

303 
279-296 

35-9 
29.3 
12.7 
42.2 
696 



III 



63 

0.689 
58.0 

50-3 
15.8 

303 
282-300 



27.9 

23.9 
15.68 

45-3 
747 



Superheated steam. 



IV 



180 
0.218 
61.0 

5°-4 
15-2 

478 
313 

27.4 
18.3 
6.83 
35-2 
631 



V 



108 

0.439 
61 .4 



50.0 
15.4 

441 
316 

13.6 
13.6 
12.37 
3!-7 
546 



VI 



75 
0.672 

59-5 

50.2 
T 5-5 

406 
3i5 

8.9 

"•5 

15-63 

35-8 

621 



A metallic thermometer or pyrometer was placed in a 
recess in the head of the cylinder. When saturated steam 
was used this pyrometer showed a large fluctuation, but when 
superheated steam was used its needle or indicator was at rest. 
Even if a part of the apparent change of temperature with satu- 
rated steam is attributed to the vibration of the needle and 
the multiplying mechanism, it is very clear that the use of 
superheated steam reduces the change of temperature of the 
cylinder-head in a remarkable manner. The effect of super- 



2,7 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

heating on the action of the cylinder-walls is also indicated 
by the per cent of water in the cylinder at cut-off and release. 
The apparent gain by comparing the amounts of steam 
used per horse-power per hour in favor of superheated steam 
is 

42.2 — 31.7 



42.2 



X 100 =25 per cent; 



but this result is of course illusive, since the superheating 
required additional coal. As the coal-consumption was not 
determined, we must compare instead the B. T. U. per horse- 
power per minute, giving a real gain of 

696 — 546 

— z X 100 = 19 per cent. 

Superheated Steam in Triple Engines. — Recent tests 
have been made in Germany on compound and triple-expan- 
sion engines using superheated steam with extraordinary 
results. In Table XXVIII are given the results of tests on 
a triple-expansion engine at Augsburg. This engine has four 
cylinders, a high-pressure, an intermediate, and two low- 
pressure cylinders. The high-pressure cylinder and one low- 
pressure cylinder are in line on one side of the fly-wheel, and 
the intermediate cylinder and the other low-pressure cylinder 
are in line on the other side. The two cranks are at right 
angles. Each cylinder is jacketed with steam on the barrel 
only and has four double-acting poppet-valves. The cylin- 
ders are cast with double walls, so that there is no chance for 
undetected leakage from the jackets. 

The steam is superheated in a coil beyond the boiler, and 
the gases afterwards pass through a feed-water heater or 
economizer, by which they are reduced to a comparatively 
low temperature. The coal used appears to have been of 
poor quality, so that while the steam- and heat-consumptions 
are good the coal used per horse-power per hour is large for 
such an engine. Comparing the best steam-consumption for 



ECONOMY OF STEAM-ENGINES. 



373 



superheated and for saturated steam, the gain from super- 
heating appears to be 



13.2 — 12 
13.2 



X 100 = 9 per cent. 



The real gain, found from a comparison of the thermal units 
per horse-power per minute, is 

235 — 228 



235 



X 100 = 3 per cent. 
Table XXVIII. 



TRIPLE-EXPANSION HORIZONTAL MILL-ENGINE AT AUGS- 
BURG WITH SUPERHEATED STEAM. 

CYLINDER DIAMETERS 27.56, 43-31, AND TWO OF 59. 06 INCHES J STROKE 

63 INCHES. 

By Professor M. Schroter, Zeitschrift des Vereines Deutscher Ingenieure, 

vol. xl, p. 249. 



Horse-power 

Duration, minutes 

Revolutions per minute 

Cut-off high-pressure cylinder. . . . 

Total expansions 

Steam-pressure at boiler, pounds 
per sq. in. above atmosphere. . . 

Barometer, pounds per sq. in 

Back-pressure, absolute, pounds 

per sq. in 

Condensation in jackets, per cent 

of total steam 

Temperatures Fahrenheit: 

At boiler, superheated 

" " saturated 

» At engine, superheated 

saturated 

Hot gases before economizer. 

" " after economizer . . 

Steam per horse-power per hour, 

pounds 

Coal per horse-power per hour, 

pounds 

Pounds of water evaporated per 

pound of coal from and at 212 F. 

B. T. U. per horse-power per min. 



Superheated. 



II 



1191 
416 

60.23 

39-9 
13-6 

89.7 
13.9 

2.0 
6.28 

433 
330- 1 
414.7 



III 



498 
288 

13.0 

3-03 

4.49 
2.44 



1167 
419 
60.10 
39-4 
13- 1 

89.4 

13.8 

1.9 

5.18 

4-11 

329-9 
418. 1 



527 
316 

12.7 

3.08 

4.29 
237 



1028 

415 
60.24 

32.7 
16.2 

89.2 
13.8 

1-7 

7.28 

448 
329-7 
420.1 



516 
300 

12.0 

2.8 D 

4.41 

2.28 



Saturated. 



IV 



1 201 
406 
60.47 
38.6 
I4.4 

90.7 
13-8 

1.9 

9.18 



330.8 
325.7 



14.3 

3.56 

3.96 
2.54 



VI 



977 

416 

60.02 

28.6 

18.9 

89.4 
13-8 

1-7 
io.57 



329.9 



326.8 



13-5 

3-65 

4.00 
2.41 



993 

420 

60.05 

28.0 

19-3 

88.9 

13-8 

1.6 
12.40 



329-5 



327.6 



13.2 

3-39 

3-86 
2-35 



374 THERMODYNAMICS OF THE STEAM-ENGINE. 

The coal used is of such inferior quality, and the con- 
sumption per horse-power per hour is so irregular that it 
cannot be used as a useful basis of comparison. 

Schmidt's System. — Table XXIX gives a resume of tests 
on various steam-engines using superheated steam according 
to a system devised by Wilhelm Schmidt. His system 
consists in part in the arrangement of the boiler for generat- 
ing the steam and in part in the construction of the engine. 
The engine and boiler on which the tests I to IV were made 
may be taken as a type of the system. 

The boiler is vertical, ten feet six inches high and five feet 
six inches in diameter. It has a corrugated fire-box and 
combustion-chamber about seven feet six inches high, from 
which the products of combustion pass through a flue 22 
inches in diameter to the superheater. The upper part of the 
furnace or combustion-chamber is crossed by two circulation- 
tubes, each about 16 inches in diameter. The fire-box just 
above the grate and the flue at the top of the combustion- 
chamber are lined with fire-brick. The superheater is a 
continuous coil of pipe, 2.4 inches in diameter, arranged in 
twelve flat coils with five turns in each. Above the super- 
heater is a feed-water heater, which is also a continuous coil 
of pipe, the diameter being 1.5 of an inch; it has four flat 
coils of five turns each. The central space inside the coils of 
the superheater and feed-water heater is occupied by a closed 
cast-iron tube 22 inches in diameter. This arrangement of 
boiler superheater and feed-water heater permits of a high 
degree of superheating together with a fair evaporative 
efficiency, as the gases from the superheater are cooled to 
about 400 F. by the feed-water heater. The superheater, 
like all coil superheaters, is subject to rapid wasting, for the 
temperature of the steam inside is about 6oo° F., while the 
gases outside are about 1200 F. In this case two such 
boilers are used to supply an engine which develops one 
hundred horse-power. 

The engine has two single-acting horizontal high-pressure 



ECONOMY OF STEAM-ENGINES. 



375 



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37 6 THERMODYNAMICS OF THE STEAM-ENGINE. 

cylinders and one double-acting vertical low-pressure cylinder. 
The high-pressure cylinders have a special form of piston-valve 
controlled by a shaft-governor; the low-pressure cylinder has 
a double-ported slide-valve. The low-pressure cylinder has 
a steam-jacket, and arrangements are made for superheating 
the steam between the high-pressure and the low-pressure 
cylinders. 

Of the tests I to IV made on this engine the first 
was made without superheating of the steam between the 
two cylinders; during the second test the steam was again 
superheated about ioo° F. above the corresponding tem- 
perature for saturated steam. The two tests III and IV were 
made at a reduced boiler-pressure. The third test was made 
with and the second without steam in the jacket of the low- 
pressure cylinder. 

The report of the test does not give the data required for 
calculating the heat-consumption, so it is necessary to depend 
on the coal-consumption per horse-power per hour in making 
comparisons with other engines, since in this case the steam- 
consumption is not a reliable guide. The coal-consumption 
is very satisfactory, but is not exceptional. 

The results of the tests VI and VII made by Professor 
Schroter on a comparatively small vertical Schmidt engine 
are exceptional in both the steam and coal economy; the 
latter, which is the only trustworthy basis of comparison, is as 
small as the best result given in Table X for the Leavitt 
pumping-engine at the Chestnut Hill station, Boston. A 
further comparison of the two engines shows that the Schmidt 
engine worked with less steam-pressure and fewer expansions. 
The Leavitt engine had 21 expansions while the Schmidt 
engine had 16 expansions in test VI and 19 in test VII; these 
last figures are not given in Table XXIX, but are given by 
Professor Schroter in his report. 

The engine tested by Professor Schroter was a vertical 
single-acting tandem compound engine, with the small cylin- 
der over the large cylinder. The high-pressure piston was a 



ECONOMY OF STEAM-ENGINES. 377 

hollow plunger long enough to carry the low-pressure piston 
at its lower end. The low-pressure piston had for its effec- 
tive area the annular ring outside the plunger. The lower 
end of the large cylinder was closed and the space under 
the large piston, including the space in the hollow plunger, 
formed an intermediate receiver with variable volume. The 
high-pressure cylinder had a piston-valve, and the low- 
pressure cylinder had a slide-valve, cored through the body 
to give a double admission after the manner of the Trick 
valve. Both valves were on one spindle and were actuated 
by a single eccentric. The boiler was of the type already 
described. 

One other test in Table XXIX is worthy of note, that is, 
test V on a compound condensing-engine which developed 69 
horse-power for 1.52 of a pound of coal per horse-power per 
hour, This, though less remarkable than the economy for 
either test VI or test VII, is exceptionally good for an engine 
of that size. Other tests of less importance are recorded in 
the same table on a vertical high-speed non-condensing engine 
and on two horizontal single-cylinder engines. The vertical 
engine was tested with saturated and with superheated steam, 
and would appear to show an enormous gain from super- 
heating; but its performance with saturated steam is too poor 
to serve as a basis of comparison. 

With the exception of tests V, VI, and VII the per- 
formance of any engine recorded in Table XXIX may be 
equalled by an engine of the same power using saturated 
steam if we take for the basis of comparison the coal con- 
sumed per horse-power per hour. The exceptionally small 
steam-consumptions shown by some tests appear to be mis- 
leading. 

Steam-jackets. — A comparison of the results of applica- 
tions of Hirn's analysis to engines with and without steam- 
jackets leads to the conclusion that the beneficial effect of a 
steam-jacket is indirect. It appears that while some heat is 
supplied by the jacket during expansion (a part of which may 



378 THERMODYNAMICS OF THE STEAM-ENGINE. 

be changed into work) the greater part is supplied during 
exhaust and is carried away by the exhaust steam and lost. 
But the indirect effect is to maintain the inner wall of the 
cylinder at a higher temperature, and so reduce the initial 
condensation, and consequently to reduce the exhaust-waste. 
It further appears that the exhaust-waste is not a proper 
criterion of the economy of the engine, and that while a 
lavish use of steam-jackets on compound engines may extin- 
guish the exhaust-waste we cannot expect to get the highest 
economy by that process. 

It is not possible from any theoretical discussion nor from 
a study of all the applications of Hirn's analysis now extant 
to determine when or to what extent steam-jackets are advan- 
tageous. We must, consequently, go directly to comparative 
tests of engines with and without steam-jackets. Properly 
we should compare engines which otherwise are identical and 
which differ only in that one is made with and the other 
without a steam-jacket. Few if any such comparisons can be 
made; in general, we must be content to compare tests made 
on' an engine with steam-jackets, some of the tests being 
made with steam supplied to the jackets and some without. 
This method of comparison is not quite fair, for it is probable 
that the engine which has a steam-jacket wastes more heat by 
radiation and conduction, both when there is and when there 
is not steam in the jacket, than does an unjacketed engine of 
the same size and power. However, if this were our only 
difficulty we should be fortunate, because the total radiation 
is in no case large for a properly lagged cylinder* 

A haphazard comparison of the numerous tests that have 
been made of engines with and without steam in the jackets 
may be made to show anything, from a most extravagant and 
improbable gain to a positive loss. It is proper, by the way, 
to say that when a jacket shows no gain the engine is prob- 
ably better and is certainly cheaper without. In order to 
arrive at any conclusion it is necessary to proceed according 
to a logical system. 



ECONOMY OF STEAM-ENGINES. 379 

In general, nothing can be learned from the comparison 
of individual tests either on a given engine with and without 
steam in the jackets or on separate engines. Either or both 
of the tests may be made under conditions which give a poor 
economy, whereas useful comparisons can be made only 
when conditions are favorable. This does not preclude a 
comparison when both tests give high economy. 

When possible, tests should be in series, extensive enough 
to determine the conditions which give the best economy 
both with and without jackets ; and in such case only the best 
results for each condition should be compared. The compari- 
son is much aided by drawing curves like those in Fig. 81, 
page 390. 

A few tests (eight or ten) may completely determine the 
interesting part of the curve, as, for example, the lowest 
curve on Fig. 81. In this case there are virtually four pairs 
of points and one more individual point which definitely 
locate the curve. The three detached points below the curve 
are for tests which differ in a minor particular. 

In all cases the comparison should be of the heat-con- 
sumption instead of the steam-consumption; for the hot water 
from the jackets can and should be returned to the boiler at 
nearly the temperature in the boiler, or else should be used 
to heat the feed-water in such a way that its heat is not 
wasted. 

Though such a method will simplify our investigations and 
in general lead to consistent results, we will find some dis- 
crepancies which cannot be reconciled, and which will show 
why there is so much difference of opinion concerning the 
advantage to be obtained from the use of steam-jackets. 

Delafond's Tests. — In 1883 an extensive and important 
investigation was made by Mons. F. Delafond on a horizontal 
Corliss engine at Creusot to determine the conditions under 
which the best economy can be obtained for such an engine. 
The engine had a steam-jacket on the barrel, but was not 
jacketed on the ends. Steam was supplied to the jacket by 



380 THERMODYNAMICS OF THE STEAM-ENGINE. 

a branch from the main steam-pipe, and the condensed water 
was drained through a steam-trap into a can, so that the 
amount of steam used in the jacket could be determined. 
The engine was tested with and without steam in the jacket, 
both condensing and non-condensing, and at various pressures 
from 35 to 100 pounds above the pressure of the atmosphere. 
The effective power and the friction of the engine were also 
obtained by aid of a friction-brake on the engine-shaft. 

The piping for the engine was so arranged that steam 
could be drawn either from a general main steam-pipe or 
from a special boiler used only during the test. Before 
making a test the engine, whi:h had been running for a 
sufficient time to come to a condition of thermal equilibrium, 
was supplied with steam from the general supply. At the 
instant for beginning the test the general supply was shut off 
and steam was taken from the special boiler during and until 
the end of the test, and then the pipe from that boiler was 
closed. The advantage of this method was that at the 
beginning and end of the test the water in the boiler was 
quiescent and its level could be accurately determined. At 
the end of a test the water-level was brought to the height 
noted at the beginning. The water required for feeding 
the special boiler during the test and for adjusting the 
water-level at the end was measured in a calibrated tank. 
As the steam-pressure in the general-supply main and in the 
special boiler was the same, there was little danger of leakage 
through the valves for controlling the steam-supply; the 
regularity and consistency of results shown by the curves of 
Figs. 79 and 80 attest to the skill and accuracy with which 
these tests were made. 

Table XXX gives the results of tests made with conden- 
sation, and Table XXXI gives the results of tests without 
condensation. All the tests both with and without conden- 
sation, but during which no steam was used in the jackets, are 
represented by the several curves of Fig. 79, while Fig. 80 
represents tests made with steam in the jackets. The curves 



ECONOMY OF STEAM-ENGINES. 38 1 

Table XXX. 

HORIZONTAL CORLISS ENGINE AT CREUSOT. 

CYLINDER DIAMETER 21. 65 INCHES ; STROKE 43. 31 INCHES | JACKET ON 
BARREL ONLY ; CONDENSING. 

By F. Delafond, Annales du Mines, 1884. 



(0 
V 

•M 



u 

<L> 

s 

3 


Q 


u 
V 

a 

U5 

c 
.2 6 

"o S 

> S3 
D S3 


Cut-off in per 
cent of stroke. 


Steam-pressure, 
pounds per sq. 


V 

"5 >> 

■S3 

- 

S3 
3 H 

(J Vh 

a 
> 


c 

S3 <J 

"* •— 


1 
u 

09 

u 



V u 

| 


Steam per horse- 
power perhour, 
pounds. 




60 
105 
75 
36 
73 
55 
80 

39 


60.O 
58.6 

59-4 
57-7 
58.8 

61.5 
59-9 
58-1 


4 
6 

9 
12. s 

5-5 

6.7 

6.7 

12.5 


96-3 
98.8 
100 
99.1 
104 
102.4 
103.8 
105.2 


27.1 
27.1 
27.0 
27.0 

27.4 
27.1 

27.4 
26.8 




109 

128.5 

161 

186 

141 

i59o 

155 

212 


23.2 






3 
4 

5 
6 

7 
8 




21.4 
22.0 
17. 1 

16.7 
16.5 
17.6 


? " 

> 

2.9 

3-2 


9 
10 
11 


120 

100 

90 

55-5 
50 
94 
102 
40 
40 


59-8 
59-3 
59-8 
58.0 

59-i 

59-6 
59-6 
59-4 
60 


7-5 

8-3 

10.5 

14 
18 

5 

5-5 
"•5 

14 


79.8 
81. 1 
80.1 

850 
84.8 
85.1 
8 3 -3 
84.1 
84.1 


27.1 
27.4 
27.1 
27. 1 
• 26.5 
27.4 
27.6 
27.1 
27.0 




126 

134 
150 
*75 
194 
112 
124 
176 
193 






21. 1 
20.8 
19.9 
20.4 

17.7 

17-3 

16.9 

*7-5 


13 

15 
16 

17 




3-o 
3-i 
1 .2 

i-5 


18 


9 1 
90 

75 
75 
3 1 
US 
Q 2 
90 

7 1 
50 


58.3 
59-5 
59-o 
58.3 
59-2 
59-9 
59 - 6 
58.8 

S9- 1 

59-o 


5-9 

9 

15-5 
22 .7 

2 5 
6 

9 
15-5 
20 

25 


60.5 

55-8 
61.2 

58.3 
61 .2 

59-9 
59-9 
60.9 
61.9 
62.3 


28.0 
27.6 
27.8 
27.6 
27.1 
27.8 
27.4 
27.1 
26.8 
26.4 




85.3 
"5 
150 
172 

186 
91.7 
117 

150 
175 

194 

75-6 

94-3 
120 
140 
165 

68.8 

95-5 
120 

J 5 2 
179 


20.4 
19. 1 
18. 1 
18.4 
18.8 


i9 

20 








23 
24 

25 
26 
27 


2-5 
2.5 
1.8 

i-5 
1.6 


18.5 
17.6 

*7-3 

17.7 

18.6 


28 


70 

80 

in 

54 
55 
98 

63 

60 

74 
50 


60.7 
58.8 
60.4 

58.8 

59-4 
60.3 

57-6 

59-7 
60.1 

59-5 


6 

9-5 
15 
21 
29 

5 
10 

14-3 
22 

29 


45 -o 
48.9 
47-9 
47-8 
47.6 
45-8 
5i-6 
49.1 
48.6 
50.2 


28.0 
28.1 
27.6 
27.6 
27. 1 
28.0 
27.6 
28.1 
27.8 
26.8 




20.7 
19.4 
18.8 
19.0 
19.8 


29 
30 
3i 
32 
33 
34 
35 
36 
37 






2.6 

2-3 

1.4 

1.4 
1.2 


19-3 
18.5 
18.2 
18.9 
19.7 


38 
39 
40 
41 
42 
43 
44 
45 


85 
68 

42-5 
20 

73 
80 
40 
25 


60.3 
61. 1 

61 .0 
60.0 
60.7 
61.9 

61. 1 
60.4 


18.2 
43 

56.7 
100 

19 

42 

58 

100 


33-i 
34-7 
36.3 
3i-7 
32.0 

33 -o 

35-i 
34-7 


27.8 
26.5 
26.0 
25.2 
27.6 
26.5 
26.0 
25.2 




106 
160 
181 
182 
in 
162 
180 
199 






22 .7 
25-3 
35-9 
19.8 
22. 1 
25-4 
33° 







1.6 
1. 1 
0.6 
0.2 



382 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table XXXI. 

HORIZONTAL CORLISS ENGINE AT CREUSOT. 

CYLINDER DIAMETER 21. 65 , INCHES | STROKE 43. 31 INCHES J JACKET ON 
BARREL ONLY; NON-CONDENSING. 

By F. Delafond, Annates des Mines, 1884. 





U 


rt.S 


SB «{ 

.2 c 

O u 
> 4) 


Cut-off in per 
cent of 
stroke. 


Steam-pressure, 
pounds per 
square inch. 


Steam used in 
jacket, per 
cent. 


u 
u 

03 c/J 

u 


Steam per 
horse-power 
per hour, 
pounds. 


1 


73 


6l.7 


13 


96.3 




147.5 


28.4 


2 


55 


6l.4 


17 


100.2 




181. 5 


26.8 


3 


25 


63.6 


20 


I02.0 


. - 


217 


25.8 


4 


80 


60.8 


II 


98.I 


2.5 


143 


22.8 


5 


60 


62.0 


13 


IO3.8 


3.4 


177.5 


22.1 


6 


36 


62.0 


16 


IO3.O 


3-1 


194 


22.4 


7 


30 


62.7 


20 


103- 5 


2.0 


237 


21.5 


8 


66 


62.0 


15.5 


73-7 




121 


27.6 


9 


60 


60.9 


18 


77.0 




136 


26.7 


10 


60 


60.0 


24-5 


76.7 




178 


24.6 


11 


30 


6O.6 


32 


77-5 




209 


24.2 


12 


70 


6l. I 


16.5 


77.0 


1-7 


137 


23-7 


13 


5o 


61.6 


23.5 


75-8 


1.2 


180 


21.8 


14 


30 


60.5 


30 


78.0 


1.3 


204 


22.0 


15 


7i 


6l.4 


24-5 


50.8 




108 


27.3 


16 


70 


6l. I 


37 


51.2 




147 


27.2 


17 


50 


6O.9 


58 


50.5 




173 


30.2 


18 


25 


60.6 


100 


34-9 




145 


46.8 


19 


70 


60.5 


23 


52.6 


1-5 


108 


25.3 


20 


60 


60.5 


34 


51.8 


I.I 


141. 5 


25.2 


21 


50 


6O.3 


58 


46.2 


0.7 


168.5 


28.7 


22 


30 


6l. I 


100 


33.7 


0.3 


147-5 


46.3 



are lettered to show the mean steam -pressure for the series 
represented and the condition, whether with or without 
condensation. Thus on Fig. 79 the lowest curve 60C repre- 
sents tests made without steam in the jackets and with con- 
densation, while the highest curve on Fig. 80 represents tests 
without steam in the jackets and without condensation, at 50 
pounds boiler-pressure. The abscissae for the curves are the 
per cents of cut-off and the ordinates are the steam-consump- 
tions in pounds per horse-power per hour. The results for 
individual tests are represented by dots, through which or near 



ECONOMY OF STEAM-ENGINES. 



383 



which the curves are drawn. As there are only a few tests in 
any series, a fair curve representing the series can be drawn 
through all the points in most cases. The exceptions are 
tests made with condensation for boiler-pressure of 80 and 




20 30 40 

Fig. 79. 



50 



60 



100 pounds per square inch. The forms of the curves SoC 
and 100C Fig. 79, were made to correspond in a general way 
to the curves 5o£7 and 60C. The discrepancies appear large 
on account of the large scale for ordinates, but they are not 
really of much importance; the largest deviation of a point 
from the curve 100C is half a pound out of about 22, which 
amounts to little more than two per cent. On Fig. 80 the 



3^4 



THERMODYNAMICS OF THE STEAM-ENGINE. 



curve SoC is drawn through the points, but though its form 
does not differ radically from the .curves 6oC and 5o£\ so 
marked a minimum at so early a cut-off is at least doubtful. 
Considering that the probable error of determining power 




from the indicator is about two per cent, it would not be 
difficult to draw an acceptable curve in place of SoC which 
should correspond to the forms of 6oC and $oC. 

The results of the four tests made with steam in the 
jacket and with condensation, and which are numbered 5, 6, 
7, and 8 in Table XXX. are represented by dots inside of 



ECONOMY OF STEaM-EAGINES. 385 

small circles on Fig. 80. It does not appear worth while to 
try to draw a curve to represent these tests. 

From a comparison of the curves on Figs. 79 and 80 repre- 
senting the results of tests stated in Tables XXX and XXXI, 
it appears that the engine when running condensing, whether 
with or without steam in the jackets, gave its best economy 
at about one-sixth cut-off. When running non-condensing 
the cut-off giving the best economy was at about one-third 
stroke. A more careful consideration of the proper cut-off 
for simple engines and of the total expansions for multiple- 
expansion engines will be given later and then somewhat fine_ 
discriminations may be attempted. The general statement 
given above will suffice for our present purpose. 

It has already been pointed out that these tests show that 
the best results are obtained from this engine running con- 
densing and without steam in the jacket for a boiler-pressure 
of 60 pounds. With steam in the jacket the advisable pres- 
sure is at least 80 pounds. Taking for comparison tests 16 
and 20 of Table XXX, the gain from the use of the jacket is 

i8„ 1 — 16.9 

- X 100 = 7 per cent. 

18. 1 r 

All the tests with steam in the jacket show a very small 
percentage condensed in the jacket, so small as to raise the 
question whether the steam-trap for removing the condensed 
water could have acted properly. But we have not the data 
for calculating the heat-consumption in any case, and so must 
rest content with the comparison made. 

The tests made on the engine without condensation are 
less complete, and on the whole it appears best to compare 
only the tests made at 75 pounds boiler-pressure. Comparing 
tests 11 and 13, Table XXXI, the gain from the use of the 
steam in the jacket is 

24.2 — 21.8 

X 100 = 10 per cent. 

24.2 



386 THERMODYNAMICS OF THE STEAM-ENGINE. 

These tests by Delafond give very good illustrations of the 
error and confusion that may arise when individual tests are 
compared to determine the advantage to be obtained from 
the use of a steam-jacket. Thus a comparison of tests 2 and 
7 of Table XXX shows an apparent gain in steam-consump- 
tion of 25 per cent, while a comparison of tests 31 and 36 
hhows a slight loss from the use of steam in the jacket. 
Both pairs of tests standing by themselves would appear to 
give a fair basis for comparison; it is only by assembling 
several series of tests, as by aid of the curve of Figs. 79 and 
80, that the real value of the steam-jacket can be determined. 

Tests on Revenue Steamers. — In 1874 three steamers, 
the Rush, the Dexter, and the Dallas, were built for the 
United States revenue marine, which were designedly alike 
in all respects except that the engines were of three distinct 
types, and the boilers were adapted to the engines. This 
was done to determine which type was best adapted to the 
needs of the service. At about the same time another 
steamer, the Gallatin, was supplied with a new engine differing 
from those in the vessels just mentioned. Again, the 
Treasury Department had a small steamer built in 1870 for the 
use of the Coast Survey, named the Bache. These five 
steamers had their engines tested under the direction of Chief 
Engineer C. H. Loring, U.S.N., and Mr. Chas. E. Emery, 
Consulting Engineer U. S. R. M. During the tests the 
vessels were secured to the dock. One test on each engine 
was long enough to determine the coal-consumption; other 
tests were shorter and for them the steam-consumption only 
was determined. The feed-water was measured in a tank with 
two compartments, which were filled and emptied alternately, 
and was used as the basis for determining the steam-con- 
sumption. 

The Rush had a compound engine with the pistons con- 
nected to cranks at right angles. The cylinders were jacketed 
on barrels and heads. The small cylinder had a separate 
cut-off valve on the back of the main valve. 



ECONOMY OF STEAM-ENGINES. 



387 



The Bache had a compound engine with both pistons on 
one continuous piston-rod, the high-pressure cylinder being 
over the low-pressure cylinder. The large cylinder only had 
a steam-jacket on barrel and ends. The engine was piped to 
run compound or single, using the low-pressure cylinder only 
in the latter condition. 

The Dexter had a single-cylinder engine without steam- 
jackets, and the Dallas had an engine of the same type, but 
intended for a lower steam-pressure. 

The Gallatin had a single-cylinder engine jacketed on the 
barrel and heads. During tests on this engine the steam- 
consumption was determined by collecting and weighing the 
water drawn from the condenser. 

The results of the tests on these several engines are given 
in Table XXXII. The best results, together with the thermal 
units per horse-power per minute, are assembled in the fol- 
lowing supplemental table: 



No. of 

test. 



Name of 
engine. 



18 
16 

13 
22 

36 
34 



Bache. . . 

Rush . . . 
Bache. . . 

Dexter . . 
Gallatin 



Method of working. 



Compound with jacket 
without " 
with " . 

Simple with jacket 

" without " .... 

" with " 

" without " .... 



en 

- • 

2% 
"3 b* 

CQ 



9-o 

CQ 



o »- 

Z o. 
Pi 



53-2 
47-7 
70.8 
53-8 
47.1 

56.5 
71 .6 

59-9 



1- c 

<U O 

11 

3 rt 

G & 
_, X 



7.0 

6.7 

6.2 

5-r 
5-3 
4-5 
3-6 
4.9 



99 
69 
266 
116 
89 
185 
179 
279 



u O 1- 

SCO 

c3 O 1) 

in 



c 

1 -a 
sf is 

~ & a 



361 
408 
327 

411 
460 

423 
361 

387 



For the present we will consider only those tests which 
may be used to determine the advantage of using steam- 
jackets; the effects of using higher pressures and of com- 
pounding will be considered later. 

Comparing tests 34 and 36 on the Gallatin with and with- 
out steam in the jacket shows a gain in steam-consumption 
from the use of the jacket of 
21.9 — 20.5 



21.9 



X 100 = 7 per cent. 



388 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table XXXII. 

TESTS ON THE U. S. REVENUE STEAMERS RUSH, DEXTER y 
DALLAS, AND GALLATIN, AND THE COAST-SURVEY 
STEAMER BACHE. 



Bache. 
Cylinder diameters, inches... 16 & 25 
Stroke, inches 24 



Ruih. 


Dexter. 


Dallas. 


Gallatin 


24 & 28 


26 


36 


34-1 


27 


36 


30 


30 



10 
V 




2 


a 


C 

.2 

•3 

a 

U 


(A 

u 

3 


c" 

O 
rt 

3 

Q 


Revolutions per 
minute. 


e/i 

C 
O 

'35 
a 
rt 

a 


H 


Boiler-pressure 
by gauge, lbs. 
per sq. in. 


Barometer, 
pounds per 
square inch. 


C/J 

.as 

„ tj 

a « 
3 a 

> 
24.O 

24-3 
24.7 


ti 

Hoe- 

a to 

U 91 b 

CQ 

3-4 
3-4 
3-3 


Percentage of 
steam used in 
jackets 


U 
% 

O 

a 

V 
a> 
1* 

O 

EC 

55-9 
77.1 
85.8 

46.4 

77-5 
99.2 
110.6 
106.0 
102.3 
134-5 

47.2 
71.8 
89.1 

54-8 

74.6 

116. 

66.7 

266. K 
168.7 

186.9 
228.1 
219 .O 
292.4 
1243 

161. 8 
196.2 


Steam per horse- 
power per hour, 
pounds. j 


I 

2 

3 


Bache, 


Comp. 
without 
jacket. 


1.83 
2.07 
2.13 

i-73 
2.07 

i-93 
1.98 
7.07 

I5-23 

2.00 


42.6 
47-7 
49-3 


9.1 
6.7 
5-6 


82.0 
80.3 
80.3 


14.7 
14.7 
14.7 




23.8 
23.0 
23.2 


4 

S 
6 

7 

8 

9 
10 


Comp. 

with 
jacket. 


38.9 
48.2 

53- 2 
56.3 
55-6 

53- 6 
60.6 


16.9 
9.2 
7.0 
5-7 
5-7 
5-* 
4.2 


81.4 
80.3 
80.2 
80.1 
80.0 
79.8 
79.1 


14.9 
14.9 
14.9 
14.9 
14-7 
14-3 
14.9 


24-5 
26.5 
26.5 
26.6 
26.1. 
24.4 
26.5 


3-7 
3-i 
3-3 
3-3 
3-4 
3-3 
3-3 


6-5 
7.0 

5-i 
4.6 

5-4 
5-6 
4.0 


25- 1 

20.7 
20.3 

20.4 
22.0 
22.4 

21.2 


11 
12 
*3 


Single 
without 
jacket. 


1.80 
1.98 
2.05 


37-3 
44.9 
47.1 


11. 8 
7-6 
5-3 


81.0 
79.6 
78.1 


14.6 
14.6 
14.6 


24.0 
23.8 
24.2 


4 .6 
4.4 
4.4 




35-o 
29.6 

C6.2 


14 

J 5 
16 


Single 

with 

jacket. 


2 . 10 
1.68 
2. 12 

1.88 


399 
46.2 

53-8 

45-3 


12.6 

8.6 

5-i 
2 2 


80.8 
81. 1 

79.6 
30.4 


14.6 
14.6 
14.6 
14.6 


24.7 
25-3 
25 5 
24.0 


3-° 

2.8 

3-7 
47 


4.0 
1.8 
2 .2 
i-5 


27. 1 

24.1 
23.2 

34-o 


18 
19 


Rush. 


Comp. 

with jacket. 


55 
6 


70.8 
55-5 


6.2 
4.0 

4-5 
3-7 
3-5 
27 

3-3 

2.4 
2. 1 


69.1 

36.7 


14.8 

14.8 


26.5 
26.2 

25-9 
25.2 

25-5 
25-3 
26.1 
26.0 

25-5 

26.1 
26.0 

25.2 

25-4 
24.8 

25-3 
25.1 

25-9 

25.1 

25-7 
25.0 


3-5 
3-4 


'} 


18.4 
22. 1 


20 
21 
22 
23 
24 
25 
26 


Dexter. 


Single 
without 
jacket. 


2.92 
1.42 

34-5 
0.65 
1.32 

1 .20 
0.92 


56.5 
64-3 
61. 1 
72.8 
So. 8 

55-3 
60.7 


68.7 

69-3 
67.1 
66.4 
40.6 

39-9 
41.9 


14.8 
14.8 
14.8 
14.8 
14.8 
14.8 
14.8 


3-4 
3-7 
3-6 
5-3 
3-2 
3-6 
4-3 

3-° 
3-4 
3-9 
4-i 
4.1 

4-3 

4-7 
3-9 

4.0 
3-6 
4-4 




23 -9 
24.1 

23 -9 
24 -3 
28.8 
28.9 
31.8 


27 
28 
29 
30 
31 


Dallas. 


Single 
without 
jacket. 


1.52 

i-55 
31.0 

1.60 
i-53 

24.0 
2.05 
2.02 

24.0 
2.22 
i-93 


48.7 
56-9 
61.5 
64-5 
63-5 


5-i 
3-4 
3-i 
2.9 

2.3 


35-4 
35-3 
32.0 

33-7 
27.4 


14.8 
14.8 

M-7 
14.8 

14.8 




138.0 
186.8 
221 .4 
242.8 
2.34-3 

247.9 

212.2 
259.O 

260.5 
179-5 
324-4 


26.7 
26.9 
26.9 
28.9 
31.0 


32 

33 
34 


Gallatin. 


Single 
without 
jacket. 


60.3 
56.0 
59-9 


4-5 
5-6 
4.9 


64. 1 
68.2 
68.5 


14.9 
14.6 
14.8 


3-5 
5-i 
3-2 


24.3 
23.8 

21 .9 ' 


35 
36 
37 


Single 

with 

jacket. 


61.5 
5i-i 
68.7 


4-5 
7-3 
4.2 


65-4 
71.6 
67.2 


147 
14.8 
14.8 


22.0 
20.5 
2I -5 



ECONOMY OF STEAM-ENGINES. 389 

A comparison of tests 13 and 16 on the Bache, using the large 
Cylinder only, shows a gain of 

26.2 — 23.2 

7 X 100 =11 per cent. 

26.2 r 

Finally, a comparison of tests 2 and 6 on the Bache as a com- 
pound engine shows a gain of 

23.0 — 20.3 

X 100 = 11 per cent 

23.0 v 

from jacketing the low-pressure cylinder only. Comparisons 
based on thermal units per horse-power per minute give the 
same results, because the per cent of steam used in the 
jackets is not large in any case. 

Experimental Engine at the Massachusetts Institute 
of Technology. — This engine, which was added to the equip- 
ment of the laboratory of steam-engineering of the Institute 
in 1890, is specially arranged for giving instruction in making 
engine-tests. It has three horizontal cylinders and two 
intermediate receivers, the piping being so arranged that any 
cylinder may be used singly or may be combined with one or 
both of the other cylinders to form a compound or a triple 
engine. Each cylinder has steam-jackets on the barrel and 
the heads, and steam may be supplied to any or all of these 
jackets at will. The steam condensed in the jackets of any 
one of the cylinders is collected under pressure in a closed 
receptacle and measured. Originally the receivers were also 
provided with steam-jackets; now they are provided with 
tubular reheaters so divided that one-third, two-thirds, or all 
the surface of the reheaters can be used. The steam con- 
densed in the reheaters is also collected and measured in a 
closed receptacle. 

The valve-gear is of the Corliss type with vacuum dash- 
pots which give a very sharp cut-off. The high-pressure and 
intermediate cylinders have only one eccentric and wrist-plate, 
and consequently cannot have a longer cut-off than half stroke 



39° 



THERMODYNAMICS OF THE STEAM-ENGINE. 



under the control of the drop cut-off mechanism. The low . 
pressure cylinder has two eccentrics and two wrist-plates, and 




10 20 30 40 

Fig. 8i. 

the admission-valves can be set to give a cut-off beyond half 

stroke. The governor is arranged to control the valves for 



ECONOMY OF STEAM-ENGINES. 39 1 

any or all of the cylinders. Each cylinder has also a hand- 
gear for controlling its valves. For experimental purposes 
the governor is set to control only the high-pressure valve- 
gear, when the engine is running compound or triple-expan- 
sion. The hand-gear is used for adjusting the cut-off for the 
other cylinder or cylinders; usually the cut-off for such 
cylinder or cylinders is set to give a very small drop between 
the cylinders. This arrangement throws a very small duty 
on the governor, so that by the aid of a large and heavy fly- 
wheel the engine can be made to give nearly identical indi- 
cator-diagrams for an entire test during which the load and 
the steam-pressure are kept constant. 

The main dimensions of the engine are as follows: 

Diameter of the high-pressure cylinder 9 inches. 

" " intermediate " . . . . c v 16 " 

" " low-pressure " 24 " 

" " piston-rods 2 T 3 ^ " 

Stroke , 30 " 

Clearance in per cent of the piston displacements: 

High-pressure cylinder, head end, 8.83 ; crank end, 9.76 
Intermediate " " " 10.4 " " 10.9 

Low-pressure '" " " 11.25 " " 8.84 

Results of tests on the engine with the cylinders arranged 
in order to form a triple-expansion engine are given in 
Table XXXIII; those on the engine with the small and 
intermediate cylinders forming a compound engine are given 
in Table XXXIV; and other tests with the tubular reheaters 
in use are given in Table XXXV. The results of tests in 
Tables XXXIII and XXXIV are represented by the diagram 
Fig. 81, with the cut-off of the high-pressure cylinder for 
abscissae and with the consumptions of thermal units per 
horse-power per minute as ordinates. 

The most important investigation which has been made 



39 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

Table XXXIII. 

TRIPLE-EXPANSION EXPERIMENTAL ENGINE AT THE MASSA- 
CHUSETTS INSTITUTE OF TECHNOLOGY. 

Irans. Am. Soc. Meek. Engs., 1892-1894; Technology Quarterly, 1896. 





G 
O 

•5 

B 
O 

u 


3 

C 

B 
i_ 

a 

en 
C 


3 

> 


1 

"So 

u 

O « 

S 3 

C 05 

OJ 0) 
U HI 

1_ u 

pu 


u 

bfl 

3 
rt 

bo 

>^ 

Xl 

U 
Ih 

3 
tn 

CO. 

u 

a 

il. 
U 

O 

PQ 


Vacuum in condenser, 
inches of mercury. 


O 

05 

OJ 

•C 
u 
a 

S£ 

u 

PQ 


Steam used in jackets, 
per cent. 


C 
u 


a 

05 

O 

X 


Steam per liorse- power 
per hour. 


u 
u 

a 

3- 3 

!r " 
a ™ 

PQ 


c 

S3 
. 3 




u 

u 

3 

ISl . 
05 U 

<u <u 
9-c 

3- 2 

2-5 
2-5 
3-2 

3-4 
3-5 
3-5 
6.1 

6-5 

5-3 
4-5 
3- 1 

2.6 
2.9 
3-o 
1.4 

3-2 

3-4 

2.9 

2. 1 
2.2 
1.4 

1-3 

1.4 

1 .2 
1 . 1 


u 
U 
> 


u 
Ih 

05 


05 

HI 

u . 
CXU 
1 V 

<U 

£ <u 

'- - 


u 
> 

'S 
u 

OJ 

u 

•a 

c 


<u 



V 

Ih 

3 
a) 

05 

<U 

V- 

a . 
s « 


V 3 
t_ 

PQ 


I 

2 

3 

4 
5 
6 

7 

8 

9 


c • 

05 <U 
-• 13 

<u % 
OJ >> 


8). 93 
90.60 
91.93 
91-55 
92-37 
84.87 

93-15 
86.70 

87-55 

84.23 
82.50 
82.13 


36 1 
35.0 
27.3 
27.0 
25.0 
21 .9 
17.4 
12.0 
8.3 


146.2 
14.7.0 
146.9 
146.7 
146.6 
1452 
146.0 
147 
146.7 


24. 1 

24.7 
24-5 
25-4 
24-5 
24.3 
26.0 
27.4 
26.0 


29.8 

3<3-3 
29.9 
30.1 

30-7 
30.1 
30.2 

30-5 
30. 1 




8.6 
8.8 

8.5 
9.8 
10.4 
11.3 
10.7 
15.2 
15-3 




6.3 

5-4 
7.2 
8.1 

10. 1 
8.7 

11. 6 

12. 2 
13 


140.8 

138.0 
125.4 
123.9 

114. 7 
105.3 
103.5 

7^-3 
67.4 


13.8 
"3-9 
13-7 
13-7 
'4-3 
14 5 
14.7 

»5-i 

16.0 


240 
241 
237 

239 
247 
250 

255 
261 

274 

253 
235 
232 

249 
244 
246 
242 

243 

256 

290 
273 
273 
269 
267 
265 
265 
267 

318 
306 
296 
287 
276 
281 
284 
280 
283 
275 
272 
278 
271 
274 
274 


233 
237 
231 
236 
240 
2 4 r 
255 
273 
274 


10 
1 1 
12 


Ditto. 


13-5 
20.5 
23.6 


145.2 
144-5 
145-3 

H3-7 
143.6 

143-2 
147.1 
145-5 
J43-7 


26.1 
26.2 
26.4 

24.7 
25.0 
25.2 
24.7 

25-5 
26 4 


30.0 
29.9 
30.1 


4-7 
6.4 
5-6 
4-7 
4-5 
6.8 


"•3 

9- 1 
8-5 




12 . 1 

9-9 
9 .8 


77.8 
101.9 
104.2 


14.7 
13-5 
13-3 


255 
237 
235 


13 

i ~ 

16 

17 

18 


§ = <"" 

--3 1- 

05 « 

*5 if .£ 


91.20 
91.40 
91.82 
91.83 
92.17 
92-57 


361 

32-8 

29.3 

27.5 

25-9 
21.9 


30.2 
30.2 
3o-5 
30.3 
3°-4 
30.6 


6.4 
7.1 
7.6 
8.9 
8.2 
7-i 


5-4 
4-3 
4-9 
3-t 
4 7 
4 1 


5-9 
6.4 
6.1 
7-3 
5-7 
77 


154-2 
145-1 
137.0 
128.8 
125.8 
120.2 


14.4 
14. 1 

14.3 
14.1 
14. 1 
14.6 


244 
240 
243 
237 
241 

2£,d 


19 
20 
21 
22 
23 
24 
25 
26 


c 
. 

«5 

_ -a 

<u « 

.a <u 
O .c 

1 — i 


84-95 
84.03 

83-35 
82.40 
81.40 
81.05 
80.28 
80.32 


9.1 
1:5.9 
15-6 
20.7 

27-3 
29.7 

34 9 
35-6 

8.4 
8-3 
10.6 
15.8 
21.3 
21.2 
21.0 
24.] 

29-5 
29. 1 
28.7 
30-7 
31.8 
35-6 
33-8 


145.8 
144.5 
144.9 

H5-3 
144.2 

143.4 
143- 1 
144.0 


25 6 

26 4 
25.6 

26.7 
24.7 
25-4 
25-5 
25.0 


30.0 
29.9 
29.8 

30 3 
29.7 
29.9 
30.2 
29.9 


7-7 

7-2 

6.8 

6.6 
7-7 
5-3 
5.0 

4.6 


8-7 

8.6 
8 
8 
5.6 
6.8 
6.4 
7-4 


55-9 
69.4 
72.8 
84.2 

97-4 
101.5 
109.4 
1 14. 1 


16 6 
'5-5 
15-5 
15-1 
15 2 
15.0 
15.0 
15.2 


285 
277 
269 
269 
26l 
263 
262 
264 


27 
28 
29 
30 
31 
3 2 
33 
34 
35 
3 b 

37 
38 

39 
40 
41 


u 






85.60 
85.62 
85.60 
84.22 

83- 3 
82.92 

82.55 
83-32 
82.67 
81.78 
82.92 
81.52 
81.57 
81 40 
81.50 


152.8 

153 -3 
152 . 1 
152.8 
152.0 
152.4 
I53-Q 
152.0 

151. 9 
152.0 

152.5 

151-5 

152.0 
152.0 
i5 T -9 


26. 1 
26. 1 
26. 1 

25-9 

26.3 

26.09 

26.02 

25.70 

25.6 

25-7 

26.0 

26. 1 

26.0 

26.04 

25-9 


29.7 

29.9 
29.8 

3°-i5 
30.0 

30.0 

29.9 
29.9 

30.1 
30.26 






1 


53-2 
55-7 
60.6 

74-9 
85.8 
86.9 
87.8 
91. 1 

99-9 
100.5 
102.4 
106.0 
108.2 
in. 2 
112.2 


] 7-3 
16.9 
16.2 
15-4 
15-1 
15 4 
15.2 

15-5 
15-5 
15-2 
15.0 
15.2 
14.9 
14 3 
!5<i 


3i8 
308 
297 
286 
277 
281 
284 
278 
280 
273 
272 
278 
271 
274 
274 



ECONOMY OF STEAM-ENGINES. 



393 



Table XXXIV. 

EXPERIMENTAL ENGINE AT THE MASSACHUSETTS INSTI- 
TUTE OF TECHNOLOGY. 

COMPOUND J CYLINDER DIAMETERS 9 AND 24 INCHES; STROKE 30 INCHES. 
Technology Quarterly , vol. xi, p. 43. 









1 u: 










V 








V 


C CU 


CU 

6 


J. C/5 W 






C/3 






a 

c/> 

C 


u 
3 

Cfl V 

CU Q£ 


O.C • 
cj >\ 

c .ss 


3 W u 


C u 
— <u 

•a 0. 


1* 

u 

5 


A* 

u a 


a=2 




._ a> 

3 3 

O.S 

cu B 


k 3 

u be 
■2>. 


c u u 

3 S2 

3 c c 


as . 


4J &X >> 

<=£ u 

<U 1) 

^'3 





Cu 

(/) 

u 


a- 

6*3 

a! 5 O 


V. 

• - u 




o-° 


c«"a 


S«« 


Job 


cu -5, 





cu 0.^3 


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M 


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> 


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CU 


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go 


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82 .72 


"9-3 


24.7 


29.56 


14.4 




5I-87 


jy-95 


359 


2 


83-25 


119. 6 


25.4 


29 80 


13-7 




54-89 


19-34 


349 


3 


82.75 


119.6 


24.9 


29.50 


16. 1 




57.00 


19.27 


347 


4 


81.88 


119. 2 


26.0 


29.50 


20.0 




63.89 


18.49 


337 


5 


81.62 


119. 6 


25-4 


29.48 


21.5 




65.36 


18.30 


33i 


6 


82.40 


121 .0 


26.4 


29.32 


20.4 




66.24 


18.07 


33i 


7 


8i.75 


"9-3 


26.1 


29.46 


23.0 




70.02 


18.17 


33 1 


8 


81.30 


119. 2 


25.1 


29.61 


24.7 




71.27 


18.14 


327 


9 


81.28 


123.9 


26.4 


29.95 


28.4 




81.41 


17.46 


317 


10 


81 .00 


125.5 


26.3 


29.74 


28.1 




81.78 


17 .62 


321 


ii 


81.38 


124.8 


26.5 


29.63 


30.0 




82.92 


17.40 


323 


12 


80.89 


119. 6 


24.7 


29.62 


33-5 




87.07 


18.01 


324 


13 


82.83 


120.6 


26.3 


30.17 


15.2 


17.4 


69.2 


17.24 


303 


14 


82.90 


120.7 


26.3 


30.00 


H-5 


15-4 


72.08 


16. 11 


284 


15 


83.20 


118. 9 


26.9 


29.40 


15.0 


16. 1 


72.61 


16.21 


289 


16 


81.48 


119. 2 


26.0 


29.50 


25.6 


12.4 


93.68 


16.24 


289 


17 


81.12 


119.5 


26.3 


29-39 


29.9 
16.3 


M-5 
19.2 


98.32 
63.07 


16.73 
15.96 


298 


18 


82.73 


100.2 


26.6 


29.49 


282 


19 


82. QO 


101 .0 


26.6 


29.49 


15-6 


20. 1 


63.19 


15-59 


275 


20 


81.72 


100.9 


26.3 


29.49 


24.0 


17-3 


78.13 


16.16 


284 


21 


81.37 


101.4 


26.3 


29.49 


25-4 


19.6 


80.47 


16.14 


284 



Table XXXV. 

TRIPLE-EXPANSION EXPERIMENTAL ENGINE AT THE MASSA- 
CHUSETTS INSTITUTE OF TECHNOLOGY WITH TUBULAR 
REHEATERS. 



Condition. 



Without 
steam in 
reheaters. 

Steam in 

first 
reheater. 



Steam in 

both 
reheaters. 





&fc 






« 1 




.. 




1 »> 


-G 


CU 




cu 


c i! 


O 


a 


of cu 
ressu 
r. 




O— • 


C . 


en 

a 
• 


3 


in c 

inc 

cury 


„ u. 
V- 3 
CU (j 


3. 3 


*<?"S 


k 3 


c >- u 

C CU cu 


a; cu 


er ce 
high 
cyhn 


u bo 

.a >> 


3 c/5 C 
3 C c 

y *)■« 

« 'a 


SB 
S 


c* 


a, 


CO 


> 


CQ 


81.8 


27 


140.7 


26.4 


30.6 


81.8 


27 


M7-5 


26. 1 


3°-4 


81.6 


29 


147.0 


25-9 


30-5 


81.2 


36 
10 


148.2 

147.2 


25-9 


30.2 


8.5-5 


25-5 


30.0 


8,3-5 


19 


146.9 


23.8 


30.2 


81.4 
85.0 


31 


146. 1 


25.8 


30.2 


8 


J 47'3 


26.6 


30.3 


84-5 


10 


146.9 


26.2 


3° -3 


82.4 


21 


147. 1 


25-3 


3°-4 


81.9 


27 


147-7 


25'4 


30.1 


82.0 


28 


146.6 


25-7 


30.2 



Per cent of 
steam used 
in reheaters. 



13 

14 





1 u 

cu 3 


X j 




S2 




u 


o~ 


3C - 

"« cu 


CU 


JS >- 









a = . 


a 


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<u 


S £ 


-!u3 


u 


rt 


:_ cu u 


O 


CU Q. 


- art 


X 


t/J 


23 


88.6 


16.0 


290 


8 7 -5 


16.0 


291 


89.7 


16.0 


290 


103.3 


15-5 


282 


66.5 


15-7 


277 


84.9 


15-9 


277 


112. 4 


15.0 


266 


61.5 


15-5 


269 


74.8 


14.9 


261 


95-7 


14.7 


258 


105.9 


14.7 


259 


107.0 


14.5 


256 



a 



a • 



k. N 

a o 



286 
281 

273 

262 
264 

274 

260 

252 

254 
254 



394 THERMODYNAMICS OF THE STEAM-ENGINE. 

on this engine is of the advantage to be obtained from the 
use of steam in the jackets. Four series of tests were made 
for this purpose: (i) with steam in all the jackets of the 
cylinders and receivers, (2) with steam in the jackets of the 
cylinders, both heads and barrels, (3) with steam in the 
jackets on the heads of the cylinders only, and (4) without 
steam in any of the jackets. 

The most economical method of running the engine was 
with steam in all the jackets on the cylinders, but without 
steam in the receiver-jackets, as shown by the lowest curve 
on Fig. 81. There is a small but distinct disadvantage from 
using steam in the receiver-jackets also. This fact could not 
be surely determined from any pair of tests, for the difference 
is not more than two per cent, and is therefore not more than 
the probable error for such a pair of tests, but a comparison 
of the two curves on Fig. 81 representing tests under the two 
conditions gives conclusive evidence with regard to this 
point. It may not be improper in this connection to call 
attention to the three points below the lowest curve and 
not connected with it; they represent tests which were made 
after the nine tests represented by points joined to the curve, 
and when some additional non-conducting covering had been 
applied to the piping and valves of the engine. Here the 
slight gain from reduced radiation is made manifest, though it 
is too small to be taken into account in making comparisons 
of the different conditions of running the engine. 

From the diagram Fig. 81 the best results with steam in 
all the jackets of the cylinders and without steam in any of 
the jackets are 233 and 274 B. T. U. per horse-power per 
minute, and the gain from the use of the steam in the jacket 
is 

274 — 233 

- f X 100 = 15 per cent. 

274 

These heat-consumptions correspond to 13.8 and 15.2 pounds 
of steam per horse-power per hour, so that on the basis of 



ECONOMY OF STEAM-ENGINES. 39 $ 

steam-consumption the gain from the use of steam in the 
jackets would appear to be only 9 per cent, instead of the 
actual gain of 15 per cent. This large difference is due to the 
large percentage of steam used in the jackets, amounting in 
all to 17 or 18 per cent of the total steam-consumption. The 
steam used in an individual jacket is, however, not excessive, 
being about 2.5 per cent in the jackets of the high-pressure 
cylinder and 7 or 8 per cent in the jackets of each of the other 
two cylinders. 

The effect of jacketing the heads of the cylinders only is 
surprisingly small, as from the diagram the best result is 262 
B. T. U. per horse-power per minute, which compared with 
the best result without steam in any of the jackets gives a 
gain of only 

274 — 262 



274 



X 100 = 4 per cent. 



The correspondence between this result and the experiments 
by Callendar and Nicolson ort the action of the cylinder-walls, 
has already been pointed out. 

The tests recorded in Table XXXIV were made on this 
same experimental engine using only the smallest and the 
largest cylinders to form a compound engine. Tests 1 to 12 
were made without steam in any of the jackets; tests 13 to 
17 were made with steam in the jackets of the low-pressure 
cylinder only; while tests 18 to 21 were made with steam in 
the jackets of both cylinders and in the receiver jacket. 
The first series of twelve tests without steam in the jackets is 
represented by the highest curve on Fig. 81. The results of 
tests with steam in the jackets are not represented on the 
diagram to avoid confusion; they are not numerous nor im- 
portant enough to warrant drawing another diagram. The 
curve representing tests without steam in the jackets does 
not show a minimum, and indicates that too large a total 
expansion for this method of running the engine was used. 
The tests with steam in all the jackets show a somewhat poorer 



39^ THERMODYNAMICS OF THE STEAM-ENGINE. 

economy than the engine has when running triple-expanding 
without steam in the jackets. We cannot draw any definite 
conclusions concerning the use of jackets from these tests. 
The evidence they present with regard to the proper ratio of 
high- and low-pressure cylinders will be considered later. 

Experimental Engine at Cornell. — The triple-expansion 
experimental engine in the laboratory of Sibley College, 
Cornell University, was built by the makers of the engine in 
the laboratory of the Massachusetts Institute of Technology 
and resembles it in all respects except that its stroke is six 
inches longer. The tests on the engine recorded in Table 
XXXVI are therefore especially valuable, as they may be 
compared directly with those made on the engine at the 
Institute. 

Tests 33 to 37 were made with the engine running triple- 
expanding without steam in any of the jackets; tests 38 to 44 
were made with steam in all the jackets of cylinders and 
receivers except the jackets on the low-pressure cylinder. 
Though differently arranged, the general effect of the jackets 
appears to be similar to that of the jackets on all three 
cylinders'of the Institute engine. The steam-pressure on the 
Cornell engine was only 115 pounds per square inch, while 
that of the Institute engine was 150 pounds. Nevertheless 
the steam- and heat-consumption for the two engines both 
with and without steam in the jackets is nearly identical, and 
the condensation of steam in the jackets under the first con- 
dition is nearly the same percentage of the total steam-con- 
sumption. Such a result is most satisfactory, as it proves 
conclusively that the tests on both engines are entirely reliable 
and free from personal error. 

The Cornell engine was also tested using the small cylin- 
der only as a simple engine, and again using the small and 
intermediate cylinders to form a compound engine. With 
both arrangements tests were made using steam in the jackets 
and without steam in the jackets. 

As a simple engine, the proportions of .the cylinder appear 



ECONOMY OF STEAM-ENGINES. 



397 



Table XXXVI. 

EXPERIMENTAL ENGINE AT CORNELL UNIVERSITY. 

CYLINDER DIAMETERS, 9, l6, AND 24 INCHES; STROKE 36 INCHES. 

Trans. Am. Soc. Mech. Engs., vol. xvi, p. 913. 





Condition. 


ej 

3 

G • 

B 

u 

V 

a 
(/> 

c 


Jj 

'0 
s> 
u 
P4 


C 
O 
'«5 

c 

a! 

a 

X 

V 



u 
4J 

8 


Steam-pressure above 
atmosphere, pounds 
per square inch. 


«4-l 

O 

XI 
<j 

£'£ 

O <u 

CQ 




(/) 

V 
Xi 



G . 
— >> 

S 3 
3 •- 
3 V 

> 


6" 

n! 
<u 
en 

.5 .J 
D C 
!_ (U 

5 u 

.22 V 

O Cu 


V 

_c 
•a . 

V) C 

3 V 

C/2 


u 
V 


a 

V 
Cfl 



X! 

■a 

<u 



-5 


u 

O in 

>- 

x°: 

»- 3 

n 

4J O. 
55 


<u w 

Q.O. 

03 


I 

2 

3 

4 
5 
6 

7 


9 X 36 

Simple without 
steam in 
jackets. 

Simple with 
steam in 
jackets. 


83.8 

84-5 
86.3 
87.0 
85.0 

87.5 
82.0 


3° 
3-o 
3-9 
5-8 
8.6 
12.0 
14.2 


95-8 

9 6 -3 
97.1 
97.1 

93-7 

100.8 

97-4 

95-7 
96.7 

97-7 
98. c 
96.9 
102.4 
99-3 

106.4 

106.0 
105.3 
107.6 

108.5 
H3-7 

106.9 
106.3 
112. 7 
112. 1 

114 3 
112. 4 

H3-4 

94-5 
95-2 
94-5 
95-4 
103.0 

114. 1 
115. 4 
114. 7 

115. 

114. 1 

116. 3 
116. 1 
114. 1 
"5-4 
114. 7 
115. 
114. i 


29.2 
29.2 
29.2 
29.2 
29.2 
29.2 
29.2 

29 • 3 
29-3 
29-3 
29.3 . 

29.3 

2Q-3 

29.3 

29-3 
29 -3 
29-3 
29'3 
29-3 
29-3 

29.4 
29.4 
29-3 
29-3 
29.3 

29-3 
29-3 

*i 4 .4 
14.5 

14.4 
14.2 

29.3 
29-3 
29-3 
29-3 
29-3 

29.3 
29-3 
29-3 
29-3 
29 -3 
29-3 
29.2 


23-5 
23.9 

23-9 
24.4 
24.7 
24.7 
24.7 


0.7 
0.7 
0.7 
0.8 
0.7 
0.9 
0.8 




73-i 
62.5 

52.4 
39 9 
30-3 
19. 1 

9-5 


25.8 

24.9 

23-5 
24.9 

25.8 

329 

47-5 

25.6 

25-5 
23.1 
23.9 
24.0 
23-4 
36.1 


. 421 


8 

9 

10 
11 
12 
13 
M 


83.6 
845 
85-4 
86.3 
87.0 
87.7 
88.2 


2-3 

3- 1 
4.4 

6-3 

9.0 

12.5 

14.2 


24.2 
24.7 
24.8 
24.9 

24.8 
24.8 
24.7 


0.7 
0.7 
0.8 
0.8 
0.7 
0.8 
0.8 


3-7 
3-9 
5-5 
6.7 
9-7 
13-4 
24.6 


71.9 
62 .4 
50.2 
40.2 

3°-3 

19.9 

y.o 


411 


15 
16 

*7 
18 

19 

20 


Compound 

without steam 

in jackets. 

16 X 36 

Compound with 

steam in h.p. 

and receiver 

jackets. 

Compound, 

steam in all 

jackets. 

Triple without 
jackets. 

Triple with 
steam in h.p. 
and int. and 

receiver 

jackets. 


83-7 
84.6 
85-6 
81.7 
87.4 
88.2 

84.4 

85.5 
86.0 
87.2 
87.7 
87.8 
88.0 


7.8 
10.2 
16.2 
26.5 
39-6 
46.4 


21.7 

21.5 
22.0 

21.5 
22.5 

22.0 

22 .6 
22.7 
21.5 
21.8 
23.8 
2 3-5 
23 -3 

*u .3 
11.6 

11. 6 
11.2 

2 3-4 
22.7 

22. 7 
22.6 
22.7 


0.8 
0.8 
0.7 
0.2 
0.0 
0.0 




97-3 
78.7 

57-7 
35-7 
24.6 
11. 4 

99.8 

77-4 
58.2 

35-6 
25.2 
20.0 
16.2 


17.6 
16.7 
16.5 
19.2 
22.8 
35-6 


292 


21 
22 

23 
24 

25 
26 
27 


7.6 
11. 4 

17.2 
32.0 
41.4 
43-1 
46.4 


3 

°-3 
0.8 

0.8 

i-3 

0.8 
0.9 


11 .1 
14.3 
14-3 
19-3 
26. 1 
29.2 
3°- 5 


18.5 
18.2 
18. s 
20.2 
21.7 
22.5 
26.2 


316 


28 
29 
3° 
3i 
3 2 


85-0 

85 3 
85.1 
85-4 
85-7 


9.0 
10. 1 
10.3 
12. 5 
13-6 


0.7 
0.7 
0.8 

0.5 
0.9 


18.9 
14.6 

15-5 
13.8 
18.0 


76.9 
71.4 
76.2 
72.4 

65-9 


18.0 

i7-3 
17.4 
16.3 

18.2 


284 


33 
34 
35 
36 
37 


84.7 

85.9 
86.9 
86.8 
88.2 


23-3 
36.6 

53-4 
62.1 
92.1 

15.8 
21.9 
29.4 

48.5 
69.0 
88.1 
96.7 






88.6 
66.1 
46. 1 

35 5 
22 .7 


»5-3 

18.0 

19 9 
24.1 

27-5 

J 5-3 
13-7 

14.9 
16.8 
17.7 
21 .1 


275 




















38 
39 
40 

41 
42 

43 
44 


83.8 
85.0 

85-5 
86.2 
•87.0 

870 
88.0 


24.2 
24-3 
233 
24.1 
22.9 
22 . 1 
21.6 






16.7 
21.8 
24.4 
27.9 
31.2 
33-8 
36.8 


141. 4 

112. 7 

89.8 

63.0 

45.6 
33-9 


237 



























* Pounds per square inch. 



398 THERMODYNAMICS OF THE STEAM-ENGINE. 

to be unfortunate, for the economy, measured either in terms 
of steam per horse-power per hour, or in terms of B. T. U. 
per horse-power per minute, is poor, even allowing for the 
low vacuum in the condenser. The condensation in the 
jackets appears to be normal, showing that the jackets were 
active, and were not using an excessive amount of steam, and 
yet the gain from using steam in the jackets is trivial, 
amounting to only 

421 — 41 1 

X 100 = 3 per cent. 

421 J r 

The detailed report by Professor Carpenter shows that 
the simple engine has nearly as much condensation and 
re-evaporation with steam in the jackets as without, except 
for tests with a very early cut-off. As the jackets did not 
have much effect on the action of the cylinder-walls they 
could not be expected to reduce the steam-consumption. 
For tests with an early cut-off, however, the jackets did 
reduce condensation and improved the economy. Thus 
tests 6 and 13 appear to show a large gain from the use 
of the jackets; but a similar gain can be obtained by 
lengthening the cut-off. This gives another illustration of 
the futility of trying to find the advantage of using steam- 
jackets from a pair of tests with and without steam in the 
jackets. 

Three series of tests were made on the engine running 
compound, using the small and intermediate cylinders. 
Tests 15 to 20 were made without steam in any of the 
jackets and tests 28 to 32 were made with steam in all the 
jackets. Again, tests 21 to 27 were made with steam in the 
jackets of the high-pressure cylinder and the intermediate 
receiver, but with much worse results than when no steam 
was used in any of the jackets. It is a question whether the 
jacketing was not overdone in both series of tests with steam 
in the jackets, for the percentage of condensation of steam in 
the jackets is large, if not excessive. If the steam-consump- 



ECONOMY OF STEAM-ENGINES. 399 

tion is taken as the basis of comparison there appears to be 
little gain from the use of steam in the jackets; for example, 
the steam-consumption for test 17 is 16.5 pounds per horse- 
power per hour, and it is 16.3 for test 31. But a comparison 
of the thermal units per horse-power per minute for these same 
tests shows a gain of 

2Q2 — 2 7^ 

-^- X 100 = 6 per cent. 

292 r 

It is probable that a better result could be obtained by 
jacketing the cylinders and not the intermediate receiver, or 
perhaps by jacketing the low-pressure cylinder only. 

Tests on Pumping- engine at Laketown. — Table 
XXXVII gives the results of tests made on a triple-expansion 
pumping-engine at Laketown, Indiana, to show the effect 
of using steam in jackets on the cylinders and in reheaters 
placed in the intermediate receiver. Taking the steam-con- 
sumption as the basis of comparison, there appears to be only 
a trivial gain from the use of steam in the jackets. But when 
the comparison is made using thermal units per horse-power 
per minute the gain is 

264 — 253 

-p X 100 = 4 per cent. 

It appears that as good a result is obtained when steam is 
used in the jackets of the intermediate and low-pressure 
cylinders as when steam is used in all the jackets and the 
reheaters. 

But the economy of this engine is poor — no better than 
that of the experimental engine at the Massachusetts Institute 
of Technology, which develops only half the power and is 
only a fifth as large. It is difficult to account for this poor 
performance, for, though the stroke is short, the valves, which 
are of the Corliss type, are in the cylinder-heads, so that the 
clearance is not excessive. 



400 THERMODYNAMICS OF THE STEAM-ENGINE. 

Table XXXVII. 

TRIPLE-EXPANSION HORIZONTAL PUMPING-ENGINE 
AT LAKETOWN, INDIANA. 

CYLINDER DIAMETERS 24^, 34, AND 54 INCHES; STROKE 36 INCHES. 

By Professor J. E. Denton, Trans. Am. Soc. Meek. Engs., vol. xiii, p. 1340. 



Duration 

Revolutions per minute 

Cut-off, h. p cylinder, per cent 

int. do. do 

1. p. do. do 

Pressures, pounds per square inch above 
atmosphere: 

Boiler 

High-pressure jacket 

Intermediate- and low - pressure 

jackets and receivers 

Vacuum, inches of mercury 

Barometer, inches of mercury 

Temperature jacket-water, degrees F. . . . 

Per cent of moisture in steam 

Percentage of condensation in jackets 

and reheaters 

Efficiency of mechanism 

Horse-power, indicated 

Steam per horse-power per hour, pounds 
B. T. U. per horse- power per minute 



Steam in all jackets and 
in reheaters. 



8 
27.8 


5 
27.6 


5 
28.3 


6 
2 7-3 


3& 


21 


22 


23 


39 


4* 


43 


40 


44 


47 


53 


47 


113 


151 


151 


151 


113 


151 


151 


I5 1 


"3 

26.2 


135 

26.1 


72 
26.2 


67 
26.1 


29.8 
336 


28.8 

350 


29.4 
308 


28.7 

300 


2.2 


2 


2 


2-5 


18 


28 


23 


1.9 


0.94 


0.92 


o.95 


0.94 


322 


323 


3 2 7 


3'7 


14-3 

260 


i3-« 

248 


14.0 

253 


J 3-7 
250 



6 
27.7 

23 

40 

47 



150 

150 



43 
26.2 

29-3 

280 

2-5 



19 

0.94 
322 

13 8 
253 



G*« 



28.0 



152 



62 
26.2 
29.1 
299 
2.4 



3 -95 
3 T 3 

14. 1 
256 



TS 



$ re 



m 



2 C 

■x re 

o u 



27.9 
23 
43 
52 



*5* 



75 
26.2 

29-3 
310 



o-93 
323 

13.8 
252 



3 
27.7 

3 1 
40 

45 



I 5 I 



26.1 
29. r 



0.91 
328 

14. 1 
264 



Gain from Using Steam-jackets. — Reviewing all the 
tests on engines with and without steam in the jackets, bear- 
ing in mind the discrepancies which have been pointed out 
and not satisfactorily explained, it appears to be conservative 
to say that from 5 to 10 per cent may be saved by jacket- 
ing simple condensing-engines and compound condensing- 
engines, and that from 10 to 15 per cent may be saved by 
jacketing triple-expansion engines, provided that these con- 
clusions shall not apply to engines of more than 300 horse- 
power. Most of the tests quoted are on engines which do not 
develop more than 150 horse-power. 

The saving on massive engines of 1000 horse-power or 



ECONOMY OF STEAM-ENGINES. 4OL 

more is likely to be smaller, and very large engines may 
derive no benefit whatever from steam-jackets. 

The saving from the use of jackets on small engines of 
five or ten horse-power may amount to 25 per cent or more. 
Isherwood found a gain of about 30 per cent from using steam 
in the jackets on an engine 5 inches in diameter by 10 
inches stroke and developing one and a half horse-power. 
Such engines are seldom if ever provided with jackets, as the 
total fuel-consumption is of little importance, and simplicity 
and low first cost are more considered than economy of 
steam-consumption. 

Intermediate Reheaters. — Many compound and triple- 
expansion engines have some method of reheating the steam 
on its way from one cylinder to another. Notable examples 
are the Leavitt pumping-engines, for which results are given in 
Table X. The r act that these engines give the best economies 
recorded for engines using saturated steam lead to the infer- 
ence that such reheaters may be used to advantage. The 
only direct evidence, however, is not so favorable, for, as has 
been pointed out on page 394, there was found a small but 
distinct disadvantage from using steam in double walls or 
jackets on the intermediate receivers of the experimental 
engine at the Massachusetts Institute of Technology. It 
appears that this engine gives the best economy when steam 
is supplied to the jackets on the cylinders and not to the 
jackets on the reheaters, and, further, that when steam is used 
in the receiver-jackets the steam in the low-pressure cylinder 
shows signs of superheating, which may be considered to 
indicate that the use of the steam-jacket is carried too far. 

After the tests referred to were finished the engine has 
been furnished with reheaters made of corrugated-copper 
tubing, so arranged that one-third, two-thirds, or all of the 
reheating-surface can be used, when desired. Table XXXV, 
page 393, gives the results of tests made on the engine with 
and without steam in the reheaters; in these tests the entire 
reheating-surface was used when steam was supplied to a 



402 THERMODYNAMICS OF THE STEAM-ENGINE. 

reheater; the effect of using part of the reheating-surface 
remains to be determined. 

For some reason the heat-consumption when no steam was 
used in the reheaters is somewhat greater than that given in 
Table XXXIII for the engine without steam in any of the 
jackets; the difference, however, is not more than two or two 
and a half per cent and cannot be considered of much impor- 
tance. It is clear from the table that there is advantage 
from using one reheater and still more from using two. If 
the heat-consumption for the engine without steam in the 
jackets and without steam in the reheaters (taken from Table 
XXXIII) is assumed to be 274 B. T. u. per minute, then the 
gain from using the reheaters appears to be 

274 — 252 

X 100 = 8 per cent, 

274 v 

which is scarcely more than half the gain from using steam in 
the jackets. These tests cannot be considered conclusive, as 
they are too few and refer only to one engine. 

Willans's Tests.— Tables XXXVIII to XLII give the 
results of a very extensive investigation by Mr. Peter Willans 
on a peculiar engine, which he designed to run at high speed 
and give a good economy. His success is shown by the fact 
that one of his engines developing only 30 horse-power used 
12.7 pounds of steam per horse-power per hour, as given in 
Table X. 

Willans's tests covered a wide range of conditions, includ- 
ing variations of steam-pressures, number of expansions, 
speed of rotation, both with and without condensation, for 
his engine when run single, compound, or triple-expanding; 
it is consequently convenient to describe his engine and 
discuss his tests before considering the effects of the various 
conditions named on the economy of steam-engines. 

The engine is a single-acting, vertical, three-cylinder, 
triple-expansion engine, with three pistons on a continuous 
piston-rod. The piston-rod is made hollow and serves as a 



ECONOMY OF STEAM-ENGINES. 



403 



Table XXXVIII. 

TESTS ON WILLANS ENGINE. 

SIMPLE NON-CONDENSING. 







3 

C 




Pressures, pou 
square inc 


rids per 
h. 




Steam per 

horse-power 

per hour, 


>> 


Per cent 

of water in 

cylinder. 




c/) 
U 

3 


e 













pou 


nds. 


c 
« 
'0 




















C 


a 

en 


u 








V 

u 
3 


u 

V 




>, 










B 


c 



u 






(A V 

^3 


5 



a 




« 9. % 


bo 

rt 





6 




4-a 


3 

> 


its j* 



1 - 


H 




v B 


•" 


<0 
en 


3 


3 u. 6£ 


c 
u 



3 



T3X 
C O 




3 


V 


3 tn 


X 


rt 


a! ™ 




<j 


O O. CL> 


<u 


*j 


j_> t/3 




° 


* 


U 
O.60 


03 


pq 


PQ 


K 


< 


P4 


Ph 


< 


< 


I 


180 


393 


14-5 


3 6 -3 


14.8 


16.5 


42.8 


34-7 


81 


12 


IO 


2 


270 


408 


O.44 


'4-5 


51.0 


14.5 


20.0 


36.0 


28.6 


80 


19 


l8 


3 


242 


409 


o-34 


14.6 


74 -o 


15-4 


25-5 


32.6 


23.0 


7 1 


27 


19 


4 


169 


403 


0.30 


14.7 


85.0 


15-4 


26.8 


29.7 


21 .2 


72 


24 


19 


5 


180 


400 


0.26 


14.8 


97.0 


15.6 


31.6 


26.9 


19.2 


72 


25 


!9 


6 


I76 


398 


0.24 


14.6 


110.0 


15-4 


3i-5 


27.8 


18.7 


67 


31 


23 


7 


298 


406 


0.22 

0.44 


14.7 
M-5 


122.0 
51.0 


15-6 
14-5 


33-6 


26.0 


17.9 


69 


30 


22 


8 


270 


408 


19.8 


36.0 


28.6 


80 


J 9 


18 


9 


J 53 


201 


0.44 


14.6 


44.0 


15.2 


9-3 


41.8 


29 2 


70 


24 


l8 


10 


122 


in 


0.44 
0.34 


14.4 
14.6 


40.2 


14.7 


6-5 


46.0 


29.4 


04 


35 


29 


11 


242 


409 


74.0 


J 5-4 


25-5 


32.6 


23.0 


7i 


27 


19 


12 


152 


205 


0.34 


14-5 


66.5 


i5-i 


14. 1 


34-4 


22.5 


t>5 


34 


23 


13 


127 


"3 


°-34 
0.26 


14.4 
14.8 


62.0 


14.7 


7-9 


40.7 


22.9 


5° 


40 


33 


14 


180 


401 


97. o- 


15-6 


31.6 


26.9 


19.2 


72 


25 


19 


15 


118 


223 


0.26 


14.7 


84.7 


15-5 


16.8 


27.8 


19.7 


71 


25 


20 


16 


178 


123 


0.26 
0.22 


15.0 
14.7 


80.0 
122.0 


15-4 
15.0 


10. 


34-1 


19.6 


58 


43 


29 


17 


298 


406 


33- 6 


26.0 


17.9 


69 


30 


21 


18 


119 


224 


0.22 


15.0 


112. 


*5-5 


20.5 


30.2 


17.7 


58 


42 


26 


!9 


123 


138 


0.22 


14.7 


105 -4 


15-5 


i3-i 


31.2 


17.7 


57 


45 


3 1 



Table XXXIX. 

TESTS ON WILLANS ENGINE. 

SIMPLE CONDENSING. 















u 
V 


Steam per 














Pressures, pounds per 


horse-power 


>> 
u 


Per cent 
of water in 




en 






sq 


uare inch. 





per hour, 


c 




3 

c 


u 

V 

a 








0. 

V 

en 


pounds. 


'0 

m 


cylinder. 




















a 


en 

c 








. . 


O 






(U 








c 


. 




u 


a .a 


'_ — ■ 
3 3 


•a 
<u 




««« 


bo 





<*< 




rt 


3 3 

E 





a 

p 

3 


.2 <U en 


"75 




H 

3 


'5 ^ iifi 


c 

<u 


3 


c 




u 

3 


qj C 


3 


enjC .3 
*_. rt 


8 as 


•a 


cr<u c 


1* 


CJ 






Q 


Pi 


u 


14.6 


< 
64.7 


pa 

1.0 




< 
25-7 


ir .2 


0- 


< 


< 


I 


139 


382 


0.5 


31.6 


44 


16 


20 


2 


no 


380 


05 


i5-i 


55-i 


1.0 


27.2 


25.2 


"S 


4b 


J 7 


15 


3 


173 


381 


°-5 


14.7 


44.8 


1.0 


21.9 


26.7 


n. 2 


42 


21 


19 


4 


162 


382 


0.5 


15-0 


35-o 


0.9 


16. 1 


28.9 


12. 1 


42 


24 


20 


5 


181 


385 


o-5 


15.0 


25.0 


0.9 


11. 5 


30.0 


12.8 


43 


25 


18 


6 


191 


378 


0.5 


15.0 


20.0 


0.8 


9.1 


29.4 


J 3-4 


46 


23 


23 


7 


128 


383 


0.29 


15. 1 


85.1 


1 .0 


33-2 


22.2 


10.3 


47 


25 


19 


8 


186 


382 


0.29 


I5-I 


75-1 


1 .0 


29.0 


23-4 


10.5 


45 


28 


23 


9 


159 


380 


0.29 


i5-i 


65.1 


0.9 


24.8 


24.0 


10.6 


44 


28 


22 


10 


130 


378 


0.29 


i5-i 


45-0 


0.8 


16.8 


26.2 


11. 


42 


33 


23 


11 


146 


382 


0.29 


i5-i 


25-1 


0.8 


9.18 


28.2 


12.0 


43 


39 


30 


12 


138 


380 


0.29 


15. 1 


20.1 


0.9 


0.9 


30-0 


13.2 


44 


39 


24 



404 THERMODYNAMICS OF THE STEAM-ENGINE 

Table XL. 

TESTS ON WILLANS ENGINE. 

COMPOUND NON-CONDENSING. 



















Steam per 


| 












V 


Point of 


Pressures, pounds 


u 


horse-power 


u 


Percent of water 






3 
C 


em- 


off. 


per square inch. 


£ 


per hour, 


3 
u 


in cylinder. 




CO 

u 
3 

c 

a 


a 

u 

a 

to 










O 
a 
■ 

CU 
CO 

u 



.3 


pounds. 




IB 

w 

O 
CD 










eu 
1* 

3 


u 

3 




u 

> <u 


u 

3 




>, 


a 


a . 


a 




c 


O 


CO u 




u 


OJ3 

a a 


CO V 

CU *-> 


•a 




.3<£i ■— 


be 
cd 


•c 53 


. 1- 
— <u 











n 73 


UT3 




rt co 


fc 3 






•*-> 


-•a 


-•a 








3 


9*e 


ac 


a 



^ 2 


^0 


a 


"rt 

3 


3 


<t! 3 


ctt 3 


w. c 




rt 


O 


•3-r 


1 ■-; 


<u 6 


l£ <0 





3 u bn 




0:3 


c — 


ttj-- 




u 


> 


tart* 


S ^ 


rz^ *-» 


w-S 


3 


&% F 


. 


i >» 


Ji s^ 


Ji >> 




3 


V 




w 


cti 


rt 


05 °S 





cu a cu 


V 


3 u 


3 U 


D (J 




Q 


& 


PC 


•J 


CQ 


n 


03 


< 


ft 


Ph 


U 


O 


X 


I 


180 


400 


0.6 


0.6 


15.0 


80.2 


15-8 


24.9 


26.2 


21 .2 


81 


5 


15 


14 


2 


123 


402 


0.6 


0.6 


iS-o 


9°-5 


15-4 


29.I 


24.2 


19.9 


82 


5 


15 


H 


3 


177 


398 


0.52 


0.6 


14.6 


90.4 


15-6 


26. 1 


24.5 


19.6 


80 


8 


] 7 


V5 


4 


118 


402 


0.52 


0.6 


14.6 


100.5 


i5-i 


30.0 


23.0 


18.6 


81 


6 


21 


17 


5 


• 3 6 4 


405 


0.47 


0.6 


14.7 


104.6 


15.0 


28.7 


23.8 


18.7 


82 


10 


16 


15 


b 


123 


403 


0.47 


0.6 


14.9 


"3-9 


15-7 


33-o 


21.4 


17.7 


83 


10 


16 


17 


7 


121 


403 


0.43 


0.6 


15.0 


113.0 


!5-7 


31.0 


21.4 


17.8 


83 


11 


T3 


17 


8 


181 


403 


0.43 


0.6 


15.0 


124.5 


15-4 


34-7 


20.8 


16.8 


81 


11 


20 


i8 


9 


124 


404 


°-39 


0.6 


14.9 


124.8 


15-8 


32.3 


21.3 


16.9 


79 


J 3 


20 


16 


IO 


189 


406 


o-39 


0.6 


14.8 


135-4 


'5-4 


3 6 -3 


20.4 


16.3 


80 


12 


19 


18 


ii 


306 


402 


0.36 


0.6 


14.7 


J33-6 


15.2 


33-3 


20.3 


16.3 


80 


14 


17 


17 


12 


184 


399 


0.36 


0.6 


14.6 


143-7 


15-4 


36.6 


20.0 


15-7 


79 


14 


20 


18 


*3 


120 


405 


o.33 


0.6 


14.4 


145.2 


15.2 


36-4 


19.7 


15.6 


79 


15 


20 


18 


14 


179 


404 


o.33 


0.6 


14.1 


i55.o 


15.0 


38.6 


19-5 


15.2 


78 


15 


21 


20 


15 
16 


3°7 
177 


402 

401 


0.31 

0.31 


0.6 








37-o 
39-6 


19.6 
19.2 






19 
*7 






0.6 


14.5 
15-0 


165.0 


14.9 


14.9 


77 


21 


20 


*7 


186 


407 


0.47 


0.6 


134.8 


15-5 


40.0 


20.8 


16.3 


79 


9 


20 


19 


18 


182 


405 


o.43 


0.6 


15-0 


i35-o 


15.0 


38.0 


20.5 


16.3 


80 


11 


20 


19 


19 


189 


406 


°-39 


0.6 


14.8 


135-4 


15-4 


36-3 


20.3 


16.3 


80 


12 


19 


18 


20 


306 


402 


0.36 


0.6 


14.7 


133-7 


i5-i 


33-3 


20.3 


id. 3 


80 


M 


17 


17 


21 


182 


403 


o.34 


0.6 


14.4 


135-° 


J5-4 


32.8 


20.0 


16.2 


81 


14 


16 


13 


22 


180 


400 


0.31 


c.6 


14.2 


i35-o 


14.8 


31.2 


20.3 


16.3 


80 


18 


19 


17 


23 


200 


404 


0.23 


0.6 


14.8 


i35«5 


15-4 


23.0 


23.1 


16.4 


7* 


25 


18 


M 


24 


123 


401 


0.60 


0.6 


14.9 


9o-5 


15-4 


29.1 


24.2 


19.9 


82 


5 


15 


14 


25 


123 


210 


0.00 


0.6 


14.8 


85.2 


15-5 


16.7 


25-3 


19-5 


77 


13 


25 


19 


26 


150 


122 


0.60 


0.6 


14.6 


83-5 


15-4 


10. 


27.0 


io-5 


72 


20 


3 1 


27 


27 


123 


402 


0.47 


0.6 


14.9 


H3-9 


15-7 


33 -o 


21.4 


17-7 


83 


10 


17 


16 


28 


125 


212 


0.47 


0.6 


14.6 


105.0 


152 


18.3 


23.1 


17.7 


7 b 


20 


27 


23 


29 


116 


124 


0.47 


0.6 


14.6 


102.6 


15.0 


11 . 1 


24.7 


17-5 
16.3 


7 1 


25 


3i 


28 


30 


189 


406 


o-39 


0.6 


14.8 


135-4 


15-4 


36.3 


20.4 


80 


12 


J 9 


18 


3i 


124 


216 


39 


0.6 


T4.6 


124.0 


15 6 


20.3 


21.3 


16.3 


77 


19 


26 


23 


32 


150 


131 


o-39 


0.6 


14.6 


120.0 


15.0 


12.7 


23-7 


16.3 


69 


3° 


33 


27 



passage for the steam to and from the cylinders. The dis- 
tribution of steam is accomplished by three single-acting 
piston-valves on one continuous valve-spindle inside the 
hollow piston-rod. The ports are cut through the side of the 
hollow piston-rod at the proper places above and below each 
piston. The cut-off is produced by allowing the admission- 
port to run into the packing-ring in the cylinder-head through 
which the piston-rod works, and is consequently very sharp. 



ECONOMY OF STEAM-ENGINES. 



405 



Table XLI. 

TESTS ON WILLANS ENGINE, COMPOUND CONDENSING. 







| 














Steam per 


| 














Point of 


Pressures, pounds 




horse-power 




Per 


:ent of water 








CUt- 


off. 


per square inch. 




per hour, 




in cylinder. 




















pounds. 














4) 


in' 


u 






<u 


s-' 





















3 


•0 

C 

">. 
O 

V 
u 

3 

U) 
09 

u 

a 

bo 


U 






3- 


U 








V- 


u 








en 
V 

3 

C 



B 
O 

Ih 


C 

a 

u 
U 

a 
tn 



_3 

"o 

> 


">> 

O 

<u 

3 

(0 

en 
u 

u 

a 


u 

<u 

<u 
S 



v 
3 
"o 

U5 

JO 
u 


O 

CO 

XI 

cS 

<u 

3 
en 
en 
eg 
u 
O. 


O 

a 

v 

en 
u 

O 

x: 

•0 
u 

rt 


3 


u 

V 

a. 

'5 5o 
0* c 


<u 
G 

cE 

V 

O 
V 

a 

c 

CD 
O 
u 


U 

T3 
C 

">> 

u 

d 
c 


13 

c 

d 

V 

tn 
e« 
<U 


<u 
•a 
a 

">» 



d 



V 

•a 
c 

">< 
u 

d 

V 
en 
cd 

V 




3 


w 












rt 


G 







U 


3 


V 


3 


V 




Q 


P6 


PC 


1— 1 


m 


pa 


P3 


< 


X 


PL. 


II 


10 


u 
28 


Oh 


I 


178 


402 


0.58 


o-45 


14.7 


134-7 


1.1 


40.I 


16.7 


9-5 


57 


18 


2 


i8q 


405 


0.58 


o-45 


14.7 


114. 7 


1.1 


33-2 


17.0 


9.8 


58 


II 


8 


28 


21 


3 


175 


401 


0.58 


0.45 


IS 1 


90.1 


1 .1 


25.6 


17-3 


10. 1 


5« 


9 


9 


30 


21 


4 


177 


404 


0.58 


o.45 


14.6 


64.6 


1 .0 


18.7 


18.0 


10.4 


58 


12 


10 


27 


17 


5 


151 


399 


0.58 


o.45 


: 4 .6 


42.6 


1 .0 


10.8 


20.3 


10.8 


53 
57 


12 
11 


10 
7 


32 


15 


6 


i?7 


311 


0.58 


o.45 


14.8 


134.8 


1 .0 


31.0 


16.3 


9-3 


29 


21 


7 


147 


311 


0.58 


o.45 


14.6 


114. 6 


0.9 


25-7 


16.9 


9.2 


55 


11 


8 


30 


23 


8 


139 


301 


0.58 


o-45 


14-5 


89-5 


0.9 


' 19-5 


17.6 


9-7 


55 


12 


9 


34 


22 


9 


184 


302 


0.58 


o-45 


15.2 


65.2 


1.1 


14.0 


18.5 


10.3 


55 


*3 


11 


36 


24 


10 


179 


300 


0.58 


0.45 


14.7 


39-7 


1 .1 


7-9 
19.9 


22.0 


11. 2 


5i 

50 


17 
12 


12 
10 


40 


21 


11 


121 


203 


0.58 


o-45 


14.6 


129.6 


0.9 


17. 1 


8.6 


38 


27 


12 


120 


198 


0.58 


45 


14.8 


89.8 


0.9 


13-3 


19.0 


0.0 


48 


18 


13 


45 


31 


1.3 


156 


203 


o.=;8 


o-45 


14.7 


64.7 


1.0 


9-4 


20.1 


9-7 


48 


18 


12 


46 


33 


14 


171 


196 


0.58 


o-45 
o.45 


14.8 


39-8 


1 .0 


5-3 


23.8 


10. y 


46 


26 


16 


48 


30 


15 


132 


115 


0.58 


14. 6 


119. 6 


1 .0 


9.0 


19.7 


8-3 


42 


19 


16 


51 


43 


16 


194 


116 


0.58 


°-45 


14.8 


89.8 


0.9 


6.7 


20. 1 


8.7 


43 


21 


16 


51 


37 


17 


170 


112 


0.58 


0.45 


14.8 


42.8 


1 . 1 
0.0 


2.9 


27.0 


10.6 


39 


32 

26 


24 


59 

32 


39 


18 


I2 3 


397 


0.28 


o.45 


14.6 


164.6 


33-2 


14.8 


11. 


57 


t 

*7 


27 


19 


175 


399 


0.28 


°-45 


14.6 


139.6 


1 .0 


27.1 


15.2 


11. 7 


57 


23 


15 


33 


25 


20 


171 


402 


28 


o-45 


14-7 


114. 7 


0.8 


. 22. 1 


15-8 


12.3 


55 


22 


ib 


3« 


26 


21 


114 


396 


0.28 


0.45 


14.8 


64 8 


0.9 


' "-7 


18.2 


12.4 


52 


32 


16 


45 


30 


22 


i55 


394 


0.28 


o.45 


14.9 

14.7 


65 -9 


8 


11. 9 


18.3 


131 


51 


28 


15 
16 


42 

37 


27 


23 


163 


296 


0.28 


o.45 


139-7 


0.9 


22.1 


15.2 


10.9 


56 


28 


28 


24 


173 


199 


0.28 


0.45 


14.7 


139-7 


0.9 


14.8 


ib- 5 


11. 4 


49 


3 1 


19 


4« 


39 


25 


166 


118 


0.28 


0.45 


14.8 


163.8 


0.9 


8.8 


17.2 


"•3 

9-9 


46 
58 


34 
3i 


25 
14 


52 
36 


43 


26 


152 


402 


0.18 


0.45 


14.9 


174.9 


0.9 


27-5 


i4-3 


25 


27 


121 


39« 


0.18 


0.45 


14.9 


J 39-9 


0.8 


21.6 


15.0 


10.3 


56 


3i 


19 


37 


27 


28 


169 


407 


0.18 


°-45 


15.0 


93-o 


8 


13.2 


17.2 


10.4 


5i 


40 
36 


18 
16 


44 
43 


3° 


29 


157 


304 


0.18 


o-45 


i5-i 


165. 1 


0.7 


19.9 


15.2 


9-7 


52 


33 


30 


i74 


300 


0.18 


0.45 


15-2 


93-2 


0.8 


10.6 


17.8 


11 .0 


49 


3« 


23 


49 


41 


3i 


*57 


203 


0.18 


0.45 


15 2 


164.2 


1 .0 


13-5 


16.8 


9-7 


47 


42 


2b 


48 


40 


32 


164 


199 


0.18 


0.45 


15-2 


80.2 


1 .0 


6.0 


21.5 


10.4 


42 


5 1 
37 


23 
16 


5^ 
35 


4 b 


33 


211 


404 


0.14 


o-45 


14.7 


186.7 


1 .2 


24.9 


14.7 


9.2 


57 


26 


34 


158 


396 


0.14 


0-4S 


14.6 


119. 6 


1.0 


16.0 


15-5 


9-4 


57 


39 


22 


39 


29 



406 



THERMODYNAMICS OF THE STEAM-ENGINE. 



H 



o 
•— i 

< 

X 

w 

cu 

I— I 

H 



r% i— i 

o 

W 



en 



O 

in 
H 

W 

H 



cu 



CO. O O" 



c 


u 


3 


c 


O 




a 


nV 




u 


c/) 


rt 


V 


3 


3 


cr 






tfl 


v- 


U 


CJ 


It 


p. 


U- 





'W 



O 3 



•japUII^D 'd 'J '3SB3J3-JI 


• 

oo oo o>CMn» • 

H M M IH M M • 


vo rovo no 
CM CM CM CM 


m o o 
ro -<a- ro 




•japuqAo *d •[ 'jio-ino 


cm o m cm >- ■<*■ • 

CM CM CM CM CM CM • 


IT) -<J- U1VO 


H 00 O 

•>*■ -^- ■<)■ 


ci ro ro 

■*■ ■*• -M- 


uapun^D "iin 'as-eaia^ 




no »n ^- co 


t^ On i/"> 
m m CM 


0»n 
CM CM CM 


•japuqAa muj 'jjo-in^ 


"J". CM NO CM M m • 

M t-l W >H H H 


t-» »» »» 

CM M 1- M 


M 04 n 


CM CM CO 


'innuijXa - d - q 'asBap'a 





CM w ON 

M M M 


■q- ■<*- t^ r-«NO 00 

W M M | M HI M 
1 


•-»pu|[AD -d -q 'jjo-ir^ 


m «r) tj- t^ o m • 


ro ■*■ r*i >o 

M H M W 


VO On On 
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ECONOMY OF STEAM-ENGINES. 407 

Steam is admitted to the upper sides of the pistons and ex- 
hausted on the down stroke; on the up stroke the steam is 
transferred to the under sides of the pistons. The spaces 
below the high-pressure and intermediate pistons have con- 
siderable volume and serve as receiver-spaces; but, as the 
volumes of these spaces increase during the up stroke and as 
work is done on the under side of the pistons on that stroke, 
each receiver-space gives an intermediate expansion, and there 
are in reality five stages of expansion for the triple engine. 
To insure a downward thrust at all times on the connecting- 
rod, an air-cushion is provided at the lower end of the hollow 
piston-rod. For this purpose, and also to provide a guide for 
the lower end of the piston-rod, there is a trunk or plunger 
working in a cylinder which is closed at the top and filled 
with air that is always at a pressure somewhat greater than 
that of the atmosphere; on the up stroke this air is com- 
pressed and it expands again on the down stroke. 

Two or three of these engines are placed side by side on 
the same bed; two engines with their cranks opposite are 
equivalent to an ordinary double-acting engine; three engines 
with their cranks at 120 form a very smoothly running com- 
bination. 

Mr. Willans first made tests in 1887 on one of his engines 
which was designed to run without condensation, and four 
years later made further tests on a condensing-engine which 
had somewhat different proportions, to accord with the greater 
range of pressures and temperatures available with a vacuum. 
The dimensions of these engines are as follows: 

Non-condensing. Condensing. 
Diameters, inches : 

High-pressure cylinder 7 6 

Intermediate cylinder 10 8.5 

Low-pressure cylinder 14 14 

Stroke, inches 6 6 

One single-acting engine was used for the tests in both 
cases to make the work simpler and easier; two or three 
engines running in combination would of co'urse do two or 



40 8 THERMODYNAMICS OF THE STEAM-ENGINE. 

three times as much work and use two or three times as much 
steam. During the tests the speed of the engine was limited 
to 400 revolutions per minute for the sake of getting clearer 
indicator-diagrams; in practice the engines tested run at 500 
revolutions per minute. 

In the tests on the non-condensing engine the steam was 
drawn from a special boiler, and the steam-consumption was 
determined by weighing the feed-water, special precautions 
being taken to have the water-level and the rate of vaporiza- 
tion in the boiler the same at the beginning and end of a 
test; Mr. Willans estimated the error in the determination of 
the steam used for a test at one fourth of one per cent. 
During the tests on the condensing engine steam v/as drawn 
from a general supply, and the exhaust-steam was condensed 
in a surface-condenser, and was collected and weighed in a 
closed tank to avoid surface evaporation. In both cases the 
steam was found, by proper tests, to be dry and saturated 
near the engine. 

By removing the upper piston and supplying steam to the 
intermediate cylinder directly, the engine was run as a com- 
pound engine; but as the steam was transferred to the space 
under the piston and did work on it before going to the low- 
pressure cylinder, the engine when running compound had 
really three stages of expansion. Again, by removing the 
two upper pistons the engine was run as a simple engine. 

The only quantities in the table which require explanation 
are those in the columns headed Steam per horse-power per 
hour required by a perfect engine and Percentage of efficiency. 
The first was calculated by a method which is nearly equiva- 
lent to that given on page 235, and the efficiency is calculated 
by dividing the steam-consumption for a non-conducting 
engine by the actual steam-consumption. Since all the steam 
passed through the cylinders of the engine, this method is 
proper for this engine. The per cent of water in the cylinder 
is a rough index of the extent of the influence of the walls of 
the cylinder. 



ECONOMY OF STEAM-ENGINES. 409 

Compounding. — The most efficacious method which has 
been devise'd to increase the amount of expansion of steam 
in an engine, and at the same time to avoid excessive cylinder- 
condensation, is compounding; that is, passing the steam in 
succession through two or more cylinders of increasing size. 
An engine with two cylinders, a small or high-pressure 
cylinder and a large or low-pressure cylinder, is called a 
compound engine. An engine with three cylinders, a high- 
pressure cylinder, an intermediate cylinder, and a low-pressure 
cylinder, is called a triple-expansion engine. A quadruple 
engine has a high-pressure cylinder, a first and a second inter- 
mediate cylinder, and a low-pressure cylinder. Any cylinder 
of a compound or multiple-expansion engine may be dupli- 
cated, that is, may be replaced by two cylinders which are 
usually of the same size. Thus, at one time a compound 
engine with one high-pressure and two low-pressure cylinders 
was much used for large steamships. Many triple engines 
have two low-pressure cylinders, which with the high-pressure 
and the intermediate cylinders make four in all. Again, some 
triple engines have two high-pressure cylinders and two low- 
pressure cylinders and one intermediate cylinder, making five 
in all. 

Two main questions are presented for investigation: (1) 
under what conditions should compounding be resorted to, 
and (2) how much gain may be expected ? The answer to 
either question will be modified by the type of engine consid- 
ered, and in only a few cases can explicit conclusions be 
drawn. The most complete investigation of these questions 
is that by Mr. Willans; our only regret is that we must hesi- 
tate to apply conclusions from tests on so small and so 
peculiar an engine to large engines of more common construc- 
tion. And yet the good economy shown by his engine under 
all conditions gives an importance to the conclusions from his 
tests which can seldom be attributed to tests on small engines. 

Taking first the tests on the non-condensing Willans 
engine, we may compare the first seven tests in Table 



4IO THERMODYNAMICS OF THE STEAM-ENGINE. 

XXXVIII, the first sixteen in Table XL, and the first seven 
in Table XLII, to determine the advantage of using a com- 
pound or a triple-expansion engine, without a vacuum. 
Mr. Willans chose the number of expansions (and the cut-off 
for the cylinder taking steam from the boiler) by a quasi- 
theoretical method, which is too intricate to be given here. 
It is enough to say that it gave a terminal pressure, in the 
cylinder exhausting into the atmosphere, of about five pounds 
above the atmosphere. One series of tests, namely, 17 to 23, 
Table XL, gives direct evidence on this matter, and shows 
that for a boiler-pressure of 135 pounds the cut-off may be 
varied from 0.31 to 0.43 of the stroke of the high-pressure 
cylinder without serious effect on the economy. For the re- 
mainder of the tests on the non-condensing engine we have 
no other direct evidence regarding the effect of cut-off, than 
the fact that the steam-consumption for any test is as good 
as the conditions would lead us to expect for an engine of 
the size, and in general the results are surprisingly good. 

The results of tests on the non-condensing engine, when 
running simple or compound, are plotted on Fig. 82, using 
the absolute boiler-pressure for abscissae and the steam-con- 
sumption for ordinates. The results of tests on the engine 
when running triple-expanding are not plotted on the diagram 
to avoid confusion, and because it does not appear necessary. 
The curve for compound tests is well located and shows con- 
clusively that little if any gain is to be expected from increas- 
ing the boiler-pressure above 180 pounds; indeed it appears 
wise to limit the pressure to 165 pounds absolute or 150 
pounds above the atmosphere. An inspection of the results of 
tests I to 7 of Table XLII shows that there is no advantage 
in increasing the boiler-pressure for the non-condensing triple 
engine above 185 pounds absolute or 170 pounds above the 
atmosphere. Neither the results of tests in Table XXXVIII 
nor the curve on Fig. 82 show conclusively the maximum 
economical steam-pressure for simple engines, but it is doubt- 
ful whether there is any real advantage in raising the steam- 



ECONOMY OF STEAM-ENGINES. 



411 



pressure above 125 pounds absolute or no pounds above the 
atmosphere. These limits of 1 10 pounds for a simple engine, 
150 pounds for a compound engine, and 170 pounds for a 
triple engine can apply only to non-condensing engines and 



40 


¥ 




• 
























36 


+ \ 














33 














. 


28 




^ 


Simple 














+ 










M 






+S. j 














>v 


v Compo 


md 














4*\4 








r20 








-f 






















16 































60 



80 



100 



120 140 
Fig. 82. 



160 



180 



200 



should be restricted to engines of the Willans type or of 
similar types. 

To determine the gain from compounding we may con- 
sider that the steam-consumption for the simple engine is 27 
pounds per horse-power per hour with a boiler-pressure of 125 
pounds absolute, that the consumption for the compound 
engine is 19.5 pounds for a boiler-pressure of 165 pounds 



412 THERMODYNAMICS OF THE STEAM-ENGINE. 

absolute, and that the consumption for the triple engine is 
18.5 pounds for a boiler-pressure of 185 pounds absolute. 
The gain from compounding is therefore 

27 — 19. 5 

^— X 100 = 28 per cent. 

27 v 

The gain from using the triple engine instead of the com- 
pound engine is 

19.5 — 18.5 

X 100 = 5 per cent. 

19.5 * D v 

Coming now to Mr. Willans' tests on his condensing 
engine, we find that for each arrangement of the engine 
{simple, compound, and triple) he made several series of tests 
with constant cut-off and varying steam-pressure for each 
series. Two such series were made for both the simple and 
the triple engine when running at full speed, and four series 
at full speed were made for the compound engine. 

As has already been pointed out, the series of simple tests 
with the cut-off at 0.5 of the stroke showed no advantage in 
raising the pressure beyond 55 pounds absolute; the series 
with the cut-off at 0.29 of the stroke showed that the steam- 
pressure could be advantageously raised to at least 85 pounds 
absolute, and under these conditions the steam-consumption 
of 22.2 pounds per horse-power per hour was probably a 
minimum. 

The engine running compound-condensing gives its best 
economy (14.3 pounds of steam per horse-power per hour) 
when running at full speed with 175 pounds absolute steam- 
pressure, and with the cut-off for the high-pressure cylinder at 
0.18 of the stroke, but nearly as good results are attained for 
a cut-off at 0.14 and for a cut-off at 0.28 of the stroke. The 
two series for the triple engine with the cut-off for the high- 
pressure cylinder at 0.3 and at 0.5 of the stroke show the 
same economy of 12,7 pounds of steam per horse-power per 
hour with the steam-pressure of 185 pounds absolute. 



ECONOMY OF STEAM-ENGINES, 413 

The gain from compounding is therefore 
22.2 — 14.3 



22.2 



X 100 =35 per cent; 



and the gain from using a triple instead of a compound engine 
is 

14.3 - 12.7 



H.3 



X 100 =11 per cent. 



It is very interesting to consider the effect of the increase 
of steam-pressure on the percentage of efficiency of the engine 
in these tests on Mr. Willans* engine. This percentage is 
greater for the non-condensing than for the condensing engine, 
and is in general greater for low than for high pressure, which 
can readily be accounted for by the greater initial condensa- 
tion and re-evaporation accompanying a wider range of pressure 
and temperature. It is, however, notable that the compound 
and triple engines maintain the efficiency better than the 
simple engine does, and that the percentage of efficiency is 
high for those engines at all pressures. The non-condensing 
engine shows about the same ratio of efficiency when running 
compound and when running triple; but the condensing 
engine has a decided advantage in this regard when running 
triple. This agrees very well with the fact that there is a 
gain of 1 1 per cent for the condensing engine, and only 5 per 
cent for the non-condensing engine, to be attained by running 
triple instead of compound. 

The tests on this engine at reduced speeds will be referred 
to later. 

Gain from Compounding. — In considering the gain to be 
attained by compounding, it has sometimes been considered 
that the engines compared should have the same steam- 
pressure, in order that the comparison should be fair to the 
simple engine; and a similar condition has been claimed for 
comparisons of compound and triple engines. But the object 
of compounding is the ability to use high pressures and large 



414 



THERMODYNAMICS OF THE STEAM-ENGINE. 



ratios of expansion to advantage. Each engine should there- 
fore be run under its best condition, using as high a pressure 
and as large an expansion as may be profitable. Of course 
each engine should have the advantage of using steam-jackets 
or superheated steam, or else both should be without these 
aids. In general, comparisons should be made only between 
engines which show a good economy for their respective 
types. 

Let us take for example tests on the experimental engine 
at Cornell University using all three cylinders to form a triple 
engine, the two smaller cylinders to form a compound engine, 
or the small cylinder as a simple engine. 

CORNELL EXPERIMENTAL ENGINE. 



Data and Results. 



Revolutions per minute , 

Steam-pressure above atmosphere 

Total expansion , 

Steam per horse-power per hour, pounds, 
B. T. U. per horse-power per minute 



Simple. 



85-4 
97-7 

3-9 
23.1 
411 



Compound. 



85.4 
95-4 
12.5 
16.3 

284 



Triple. 



85.O 

Il6.I 

21.9 

13.7 

237 



It is probable that the simple engine has too high steam- 
pressure and the triple engine has too low pressure. How- 
ever, making comparisons as the results stand, we have as the 
gain from compounding 

411 — 284 



411 



X 100 = 31 per cent, 



and from using a triple instead of a compound engine 

284— 237 



284 



X 100 = 20 per cent. 



On the other hand, a comparison of the results of tests on 
the engines of the Rush and the Gallatin (Table XXXII, 
page 388, tests 18 and 36) shows a gain of only 

20.5 — 18.4 



20.5 



X 100 = 10 per cent; 



ECONOMY OF STEAM-ENGINES. 



415 



but it is very clear that the steam-pressure and the expansion 
are too small for the best economy of a compound engine. 
And again, while the Gallatin shows a fair economy for a 
simple engine, the economy of the Rush is poor. 

Taking now the best results of tests on simple, compound, 
and triple engines from Table X, all being supplied with 
steam-jackets, and the compound and triple engines with 
intermediate reheaters, we have the following results : 



Data and Results. 



Revolutions per minute 

Steam-pressure above atmosphere, pounds. 

Total expansion 

Steam per horse-power per hour, pounds. . 
B. T. U. per horse-power per minute 



Simple 
Corliss at 
Creusot. 



60 

84 

9 
16.9 



Compound L ™P ,e 

Leavittat p V ™„r 

t n„ Chestnut 

Louisville. tj:i] 



18.6 

137 

20 

12.2 
222 



50.6 
I76 

21 

II. 2 
204 



Using the steam-consumption as the basis of comparison, 
we have for the gain from compounding 



16.9 — 12.2 
16.9 



X 100 = 28 per cent; 



and for the gain from using a triple instead of a compound 
engine 

12.2 — 11. 2 



12.2 



X 100 = 8 per cent. 



The total expansion for the two Leavitt engines is nearly 
the same, but is obtained by different means. The ratio of the 
large to the small cylinder of the compound engine is a trifle 
less than four, and the cut-off for the high-pressure cylinder 
is a little less than one-fifth stroke. The triple engine has a 
little more than eight for the extreme ratio of the cylinders, 
and has the cut-off for the high-pressure cylinder at a little 
more than four tenths. In the design of a compound engine 
the desired expansion may be attained either by using a small 
ratio for the cylinders and a short cut-off for the high-pressure 



41 6 THERMODYNAMICS OF THE STEAM-ENGINE. 

cylinder, or by using a large ratio for the cylinders and a long 
cut-off on the high-pressure cylinder. The compound pump- 
ing-engine at Louisville now under discussion was designed 
by the first method, giving remarkable results. Some de- 
signers have, on the other hand, claimed an exceptional econ- 
omy for certain compound engines which have a large ratio of 
large to small cylinders, and have preferred them to triple 
engines. In confirmation we have the results of tests in 
Table XX, page 362, on an engine which was run triple, using 
three cylinders, and compound, using only the large and small 
cylinders, omitting the intermediate cylinder. These tests 
show as good an economy for the compound engine as for the 
triple engine, and tests given in Table XXIII on an engine 
with a larger ratio of cylinders and a very large total expan- 
sion show even better economy. But the result (13 pounds 
of steam per horse-power per hour), though good, is not 
so low as that for the Louisville engine, and is but little 
better than is given in Table XIX for a compound mill-engine 
at New Bedford, which has a moderate ratio of cylinders and 
much smaller expansion. The New Bedford engine was sup- 
plied with slightly superheated steam, but had no steam- 
jackets. Finally the indicator diagrams from the high-pressure 
cylinder of the Holyoke engine (Table XX) when running 
triple showed clearly that the cut-off for the intermediate 
cylinder was too short and the back-pressure for the high- 
pressure cylinder was too high, because the expansion line 
reached the back-pressure line at about two-thirds stroke. It 
cannot therefore be determined which method of obtaining 
the desired expansion for a compound engine is the better. 

Marine engines will not run smoothly with a short cut-off, 
and consequently a large number of expansions can be 
obtained only by using a large ratio of cylinders. But even 
for triple engines the ratio of the large to the small cylinder 
is commonly five or six and the total expansions are corre- 
spondingly small. The advantage of using a larger ratio, 
when allowable, is shown by comparing the tests in Table 



ECONOMY OF STEAM-ENGINES. 417 

XIV, page 358, on the Meteor and on the Iona ; the engine 
of the Iona had the further advantage of higher steam-pressure 
and a relatively early cut-off on the high-pressure cylinder. 

In conclusion, it may be said that for condensing engines 
there is no advantage in using more than 80 pounds steam- 
pressure, while compound engines may advantageously have 
the pressure raised to 135 pounds above the atmosphere. 
The gain from higher steam-pressure and compounding will 
be 25 to 30 per cent. The best pressure for triple engines 
cannot now be determined from experiments; it is, however, 
doubtful if there is any advantage in using more than 175 
pounds above the atmosphere. Such a further increase of 
pressure and the use of a triple instead of a compound engine 
may be expected to give 8 or 10 percent better economy. 

For a simple non-condensing engine the steam-pressure 
may be 100 to 115 pounds above the atmosphere, and for a 
compound engine the pressure may be 150 pounds, while for 
a triple engine the pressure may be 175 pounds or possibly 
somewhat more. The gain from compounding will be 20 to 
30 per cent, and the gain from using a triple instead of a 
compound engine will be 5 per cent or perhaps a little more. 

These conclusions apply only when the engine is run at 
full power and at the best point of cut-off or the most 
economical total expansion. In general the compound engine 
will suffer more loss of economy when running at a reduced 
load than a simple engine will; and a triple engine will suffer 
even more than a compound engine from the same cause. 

Cut-off and Expansion. — It has already been pointed out 
on page 385 in connection with Delafond's tests that the best 
point of cut-off for a simple engine, whether jacketed or not, 
is about one-third stroke when the engine is non-condensing 
and it is about one-sixth stroke when condensing. In general, 
other tests on simple engines such as those on the BacJie, 
Dexter, and Gallatin (Table XXXII, page 388), and on the 
small Corliss engine at the Massachusetts Institute of Tech- 
nology (Tables II and XXVII, pages 318 and 371), confirm 



41 8 THERMODYNAMICS OF THE STEAM-ENGINE. 

these conclusions. Tests on the experimental engine at 
Cornell (Table XXXVI, page 397) indicate that a longer cut- 
off than one-sixth may be used to advantage, but these tests 
were made with a poor vacuum and consequently show a poor 
economy for all grades of cut-off. 

The term total expansion for a compound or a triple 
engine can properly have only a conventional significance; it 
is usually taken to be the product of the ratio of the large to 
the small cylinder by the reciprocal of the fraction of the 
stroke at cut-off for the high-pressure cylinder. This conven- 
tional total expansion is about 20 for all the tests on triple 
engines quoted in Table X, page 354, except those on marine 
engines, which show a relatively poor economy. It may there- 
fore be concluded that it is not advisable to use much more 
expansion for any triple engine, and that less expansion 
should be used only when the conditions of service (for exam- 
ple, at sea) prevent the use of large expansion. 

The case is not so clear for compound engines. The 
Louisville engine, which gives the best economy of any com- 
pound engine quoted in Table X, has 20 expansions. The 
Natick engine has 33 expansions, which are clearly excessive, 
and the engine at New Bedford (page 361) has only 13.4. 
The Cornell engine (page 397) gives its best economy when 
running compound with 12 or 16 expansions. We may con- 
clude that in general 15 expansions may be used to advantage; 
marine engines and other engines which cannot readily be 
designed for so much expansion may be expected to give a 
less economy in consequence. 

Variation of Load. — In general an engine should be so 
designed that it may give a fair economy for a considerable 
range of load or power. Very commonly the engine will have 
sufficient range of power with good economy if designed to give 
the best economy at the normal load. In general, however, 
it is well to assign a less expansion and consequently a longer 
cut-off to the engine than would be determined from a con- 
sideration of the steam- (or heat-) consumption alone. For, 



ECONOMY OF STEAM-ENGINES. 419 

in the first place, the best brake or dynamic economy is always 
attained for a little longer cut-off than that which gives the 
best indicated economy, and in the second place the econ- 
omy is less affected by lengthening than by shortening the 
cut-off. The first comes from the fact that the frictional 
losses of the engine increase less rapidly than the power, as 
will be shown in the next chapter; and the second is evident 
from consideration of curves of steam-consumption as given 
by Fig. 81, page 390, and Figs. 79 and 80, pages 383-4. 

The allowable range of power for a simple engine is greater 
than for a compound or a triple engine. Comparisons for a 
simple and a triple engine may be made by aid of Figs. 
80 and 81. The Corliss engine at Creusot when supplied 
with steam at 60 pounds pressure, with condensation and with 
steam in the jacket, developed 150 horse-power and used 17.3 
pounds of steam per horse-power per hour. If the increase 
be limited to 10 per cent of the best economy, that is, to 19 
pounds per horse-power per hour, the horse-power may be 
reduced to about 92, giving a reduction of nearly 40 per cent 
from the normal power. The triple engine at the Massa- 
chusetts Institute of Technology with steam at 150 pounds 
pressure and using steam in all the cylinder-jackets developed 
140 horse-power and used 233 B. T. U. per horse-power per 
minute. Again, limiting the increased consumption to 10 per 
cent or to 254 B. T. U., the power may be reduced to about 
104 horse-power, giving a reduction of 26 per cent from the 
normal power. The effect of increasing power for these 
engines cannot be well shown from the tests made on them, 
but there is reason to believe that the simple engine would 
preserve its advantage if a comparison could be made. 
Though the tests which we have on compound engines do not 
allow us to make a similar investigation of the effect of 
changing load, there is no doubt that it is intermediate in this 
respect between the simple and the triple engine. 

When the power developed by a compound engine is 
reduced by shortening the cut-off of the high-pressure cylin- 



420 THERMOD YNAMICS OF THE STEAM-ENGINE. 

der, the cut-off of the low-pressure cylinder must be shortened 
at the same time to preserve a proper distribution of power 
and division of the range of temperature between the cylin- 
ders. If this is not done the work will be developed mainly 
in the high-pressure cylinder, which will be subjected to a 
large fluctuation of temperature, and the engine will lose the 
advantages sought from compounding. A compound non- 
condensing engine, if the cut-off for the large cylinder is fixed, 
is likely to have a loop on the low-pressure indicator-diagram 
due to expansion below the atmosphere, if the power is 
reduced by shortening the cut-off of the high-pressure cyl- 
inder. Such a loop is always accompanied by a large loss of 
economy; if the loop is large the engine may be more waste- 
ful than a simple engine, for the high-pressure piston develops 
nearly all the power and may have to drag the low-pressure 
piston, which is then worse than useless. 

There is seldom much difficulty in running a simple engine 
at any desired reduced power by shortening the cut-off or 
reducing the steam-pressure, or by a combination of the two 
methods. But a compound engine sometimes gives trouble 
when run at very low power (even when attention is given to 
the cut-off of the low-pressure cylinder), which usually takes 
the form just discussed; i.e., the power is developed mainly 
in the high-pressure cylinder. Triple engines are even more 
troublesome in this way. A compound or triple engine 
running at much reduced power is subject not only to loss of 
economy and to irregular action, but the inside surface of the 
low-pressure cylinder is liable to be cut or abraded. 

Automatic and Throttle Engines. — The power of an 
engine may be regulated by (i) controlling the steam-pressure 
or (2) by adjusting the cut-off. Usually these two methods 
are used separately, but in some instances they are used in 
combination. Thus a locomotive-driver may reduce the power 
of his engine either by shortening the cut-off or by partially 
closing the throttle-valve, or he may do both at once. Sta- 
tionary engines are usually run at a fixed speed and are con- 



ECONOMY OF STEAM-ENGINES. 4 21 

trolled by mechanical governors, which commonly consist of 
revolving weights that are urged away from the axis of revo- 
lution by centrifugal force and are restrained by the attrac- 
tion of gravity or by the tension of springs. 

The earliest and simplest steam-engine governor, invented 
by Watt, has a pair of revolving pendulums (balls on the ends 
of rods that are hinged to a vertical spindle at their upper 
ends) which are urged out by centrifugal force and are drawn 
down by gravity. When the engine is running steadily at a 
given speed the forces acting on the governor are in equilib- 
rium and the balls revolve in a certain horizontal plane. If 
the load on the engine is reduced the engine speeds up and the 
balls move outward and upward until a new position of equili- 
brium is found with the balls revolving in a higher horizontal 
plane. Through a proper system of links and levers the up- 
ward motion of the balls is made to partially close a throttle- 
valve in the pipe which supplies steam to the engine and thus 
adjusts the work of the engine to the load. 

Shaft-governors have large revolving-weights whose centri- 
fugal forces are balanced by strong springs. They are powerful 
enough to control the distribution or the cut-off valve of the 
engine, which, however, must be balanced so that it may move 
easily. 

Automatic engines, like the Corliss engines, have four 
valves, two for admission and two for exhaust of steam. The 
admission, release, and compression are fixed, but the cut-off 
is controlled by the governor. Usually an admission-valve is 
attached to the actuating mechanism by a latch or similar 
device, which can be opened by the governor, and then the 
valve is closed by gravity, by a spring, or by some other inde- 
pendent device. The office of the governor is to control the 
position of a stop against which the latch strikes and by which 
it is opened to release the valve. 

Corliss and other automatic engines have long had a 
deserved reputation for economy, which is commonly attribu- 



422 THERMODYNAMICS OF THE STEAM-ENGINE. 

ted to their method of regulation. It is true that the valve- 
gears of such engines are adapted to give an early cut-off, which 
is one of the elements of the design of an economical simple 
engine, but their advantage over some other engines is to be 
largely attributed to the small clearance which the use of four 
valves makes convenient, and to the facf that the exhaust- 
steam is led immediately away from the engine, without hav- 
ing a chance to abstract heat after it leaves the cylinder. 
These engines also are free from the loss which Callendar and 
Nicolson attribute to direct leakage from the steam to the ex- 
haust side of slide-valves, and to valves of similar construction. 
And yet the Hoadley engine (Table XXII, page 363), which 
was of the second type having a piston-valve controlled by a 
shaft-governor, compares very favorably with the Corliss en- 
gine at Creusot (Table XXXI, page 382), though it must be 
admitted that the performance of the Hoadley engine is 
exceptionally good for its type. 

Every steam-engine should have a reserve of power in ex- 
cess of its normal power; and again it is convenient if not 
essential that a single-cylinder engine should be able to carry 
steam through the greater part of its stroke in starting. 
These conditions, together with the fact that it is somewhat 
difficult to design a plain slide-valve engine to give an early 
cut-off, have led to the use of a long cut-off for engines con- 
trolled by a throttle-governor. The tests on the Corliss en- 
gine at Creusot (Tables XXX and XXXI, pp. 381 and 382) 
show clearly the disadvantage of using a long cut-off for sim- 
ple engines. It has already beer; pointed out that a non-con- 
densing engine should have the cut-off at about one-third 
stroke. With cut-off at that point and with 75 lbs. steam- 
pressure the engine developed 209 horse-power and used 24.2 
lbs. of steam per horse-power per hour when running without 
steam in the jacket and without condensation. If the steam- 
pressure is reduced to 50 lbs. and the cut-off is lengthened to 
58 per cent of the stroke, the steam-consumption is increased 



ECONOMY OF STEAM-ENGINES. 4 2 3 

to 30.2 lbs. per horse-power per hour, the horse-power being 
then 173. The gain from using the shorter cut-off is 

30.2 — 24.2 

X 100 = 20 per cent. 

30.2 F 

A similar comparison for the same engine running with a 
vacuum and with steam in the jacket shows even a larger dif- 
ference. Thus in test 16 the steam-pressure is 84 lbs. and 
the cut-off is at 11.5 per cent of the stroke, the horse-power 
is 176, and the steam-consumption per horse-power per hour 
is 16.9 lbs., while the consumption for about the same power 
in test 44 is 25.4 lbs. of steam per horse-power per hour, the 
steam-pressure being 35 and the cut-off at 58 perx nt of the 
stroke ; here the gain from using the shorter cut-off is 

25.4 — 16. 9 

— - x 100 = 33 per ceiit. 

25.4 JJ ^ 

Considering also that automatic engines are usually well 
built and carefully attended to, while throttling-engines are 
often cheaply built and neglected, the good reputation of the 
one and the bad reputation of the other are easily accounted 
for. 

It is, however, far from certain that an automatic engine 
will have a decided advantage over a throttle-engine, provided 
the latter is skilfully designed, well built and cared for, and 
arranged to run at the proper cut-off. Considering the rapid 
increase in steam-consumption per horse-power per hour when 
the cut-off is unduly shortened, it is not unreasonable to ex- 
pect as good if not better results from a simple throttling- 
engine than from an automatic engine when both are run for 
a large part of the time at reduced power. 

The disadvantage of running a compound or a triple en- 
gine with too little expansion can be seen by comparing the 
two tests on the engine of the Rusk (Table XXXII, p. 388), 
on the other hand the great disadvantage of too much expan- 
sion is made evident from the tests on the engine in the lab- 



424 THERMODYNAMICS OF THE STEAM-ENGINE. 

oratory of the Massachusetts Institute of Technology (Table 
XXXIII, p. 392). Considering that the allowable variation 
from the most economical cut-off is more limited for a com- 
pound or a triple engine, it appears that there is less reason 
for using an automatic governor instead of a throttling gov- 
ernor for compound and triple engines than there is with sim- 
ple engines. Nevertheless the most economical engines (sim- 
ple, compound, or triple) are automatic engines. 

Effect of Speed on Condensation. — Though the con- 
densation of steam on the walls of the cylinder of an engine 
is very rapid, it is not instantaneous. It appears reasonable 
that the amount of condensation, and consequently the in- 
fluence of the cylinder-walls, may be reduced by increasing 
the speed of revolution. This conclusion is confirmed by the 
results of the investigations by Callendar and Nicolson, 
and also by the tests on the Willans engine. Table IX, 
page 349, shows that the condensation per stroke was re- 
duced from 0.0159 to 0.0074 by increasing the revolutions 
from 46 to 97 per minute. It must be borne in mind in 
this connection that the effect of external radiation and con- 
vection is nearly proportional to the time, and that, conse- 
quently, its influence is more pronounced at low speeds than 
at high speeds. Since the engine used by Callendar and 
Nicolson was run at relatively low speeds and small powers, 
the influence of external radiation and conduction must have 
been abnormally large, and consequently an increase of speed 
had notable effect in the manner just explained. 

Table XXXVIII, page 403, gives four sets of three tests 
each on the simple non-condensing Willans engine ; the three 
tests of a set were intended to be made at 100, 200, and 400 
revolutions, all other conditions for the set being the same. 
Each set shows a notable gain in economy and a notable 
reduction in condensation on account of the increase of speed. 
Thus a comparison of tests 8 and 10 shows that the con- 
densation is reduced by nearly one half, and that there is a 
gain of more than 20 per cent in steam-consumption, by in- 



ECONOMY OF STEAM-ENGINES. 425 

creasing the speed from 1 1 1 to 408 revolutions per minute. 
The other sets of tests show similar results, as do also three 
sets of speed tests at the bottom of Table XL on the com- 
pound non-condensing engine. 

In the development of applications of electricity there has 
been a tendency to use high-speed engines, which can either 
be connected directly to the electric generators, or connected 
to them by belting without an intermediate shaft. In general 
the engines used have consumed more steam per horse-power 
per hour than slow-speed engines running under similar con- 
ditions. This result may be attributed to the type of engine, 
which is virtually a plain slide-valve engine controlled by a 
shaft-governor. In order that the valve may be controlled 
directly by the governor it must move freely, and con- 
sequently some form of balanced valve or piston-valve must 
be used. Such valves may be tight when properly adjusted 
and in good condition, but they are likely to leak in ordinary 
service. And again, in order that the engine may run with a 
comparatively early cut-off, the engine must have a large 
clearance or else the compression will be excessive. Com- 
pound high-speed engines have been developed to meet the 
requirements of this service, and have given satisfaction when 
run at full power. At reduced power certain difficulties arise, 
as stated on page 420, especially when the cut-off for the 
low-pressure cylinder is fixed. 

Steam-turbines. — Many attempts have been made from 
time to time to devise some form of impulse or reaction wheel 
that can be driven by the direct action of steam. The funda- 
mental difficulty in devising such a wheel is the high velocity 
of steam when flowing out of an orifice, for the linear velocity 
of the wheel must have a proper relation to the velocity of 
the steam in order to obtain economical results. 

Fig. 83 represents the essential parts of a Laval steam- 
turbine, consisting of a wheel which carries a large number of 
curved steel cups or buckets to receive the jet of steam from 
a nozzle set at an angle. The figure shows only one nozzle, 



426 



THERMODYNAMICS OF THE STEAM-ENGINE. 



but a number may be used, the wheel in question having four. 
Three of the nozzles have a diameter of 0.138 of an inch at 
the throat and one a diameter of 0.157 °f an inch. All the 
nozzles are diverging toward the exit end and are intended to 
give an adiabatic expansion of the steam to the pressure of 
the atmosphere, so as to develop the maximum kinetic energy 




PLAN 



Fig. 83. 

at the exit. The wheel runs at approximately 24,000 revolu- 
tions per minute, and as it has a diameter of about five inches 
at the middle of the buckets, their velocity is about 30,000 
feet per minute or 500 feet per second. The velocity of steam 
flowing from a pressure of 125 pounds by the gauge or 140 
pounds absolute, with continuous expansion to the pressure 
of the atmosphere, is about 2800 feet per second. 

The results of tests on this turbine by Professor Goss are 
given in Table XLIII. It is reported that a test of a 64 
horse-power Laval turbine by Prof. J. E. Cederblom gave a 
steam-consumption of 19.7 pounds per brake horse-power per 
hour when supplied with steam at about 100 pounds by 
the gauge, and exhausting into a vacuum of 26 inches of 
mercury. 



ECONOMY OF STEAM-ENGINES. 



427 



Table XLIII. 

TESTS ON A LAVAL STEAM-TURBINE. 
By Prof. W. F. M. Goss, Trans. Am. Soc. Mech. Engs., vol. xvii, p. 81. 







u 
V 

a 
</] 

c 

■sji 

o.S 


u 

V 


a 

V 

to 

u 


X! 

V 
M 

cd 

u 


Steam-pressure 
by gauge. 


u 
zs 
c 

X! 

u 




'0 

.O 

< 


V 

c 

'So 
c 
u 

u 
rt 


O- 1 ^3 
DC 

E " 3 
« ° 

t/3 


Four nozzles in action, three hav- 
ing a diameter of 0.138 of an inch 
and one 0.157 of an inch. 


1 
2 

3 
4 
5 
6 

7 
8 

9 
10 


21180 
25210 
20190 
20980 
18990 
20520 
21080 
25520 
24300 
23880 


O.OO 
I.63 
2.36 

2-97 
3-46 
4-38 
5-IO 

7-52 
8.24 

10.33 


I30 


17. 1 
42.2 

48.5 

55-6 
61.9 
70.8 
76.9 
99.6 
104.4 
126.3 


128.6 
99.8 

85.7 
79.6 

71-5 
64.4 

53-6 
5i-3 
47-8 


Three nozzles in action, two 0.138 
of an inch and one 0.157 of an 
inch in diameter. 


11 

12 
13 
14 


17960 
20920 
21050 
24660 


0.00 
3-95 
4-77 
6.50 




3i-3 
83.6 

93-4 
in. 7 


67.8 
60.0 

53-3 


Two nozzles in use, each 0.138 of 
an inch in diameter. 


*5 

16 

17 

18 


25220 
20300 
18920 
23900 


0.00 
i-95 

3-43 
3-87 




42.2 

83-5 
121. 1 
127.0 


83-4 
65.0 

59-3 



The Parsons steam-turbine has a number of turbines 
arranged in series through which the steam passes in succes- 
sion, thus breaking up the difference of pressure between the 
supply and the exhaust into a number of steps; consequently 
the velocity of the steam impinging onto any set of vanes is 
comparatively low. Instead of a few nozzles, each turbine 
has a series of guides extending entirely around the wheel, 
thus supplying steam to all the vanes of the movable wheels 
at the same time. Table XLIV gives the results of tests on 
a Parsons turbine having seven separate turbines and seven 
corresponding steps in the reduction of the steam-pressure 
from the supply to the exhaust; the low-pressure turbine was 



428 



THERMODYNAMICS OF THE STEAM-ENGINE. 



made double, thus giving eight moving wheels in all. Later 
turbines have been given a large number of separate turbines 
so that a large number of expansions of steam is attained, and 
it is claimed that their economy is as good as that of the best 
steam-engines. 

Table XLIV. 

TESTS ON PARSON'S STEAM-TURBINE. 

SIX HIGH-PRESSURE SINGLE DISKS J ONE LOW-PRESSURE DOUBLE DISK. 

By Prof. Alex. B. W. Kennedy, Engineering , vol. lvi. p. 126. 



Duration, minutes 

Steam-pressure, pounds per square inch above 

atmosphere 

Corresponding temperature of saturated steam. 

Temperature of steam near engine 

Vacuum, pounds per square inch 

Revolutions per minute 

Kilowatts 

Electrical horse-power 

Steam per kilowatt per hour, pounds 

Steam per electrical horse-power per hour, 

pounds , . .' 



72 

97 

336 

356 

14-3 

4480 

27.9 

37-5 
44.1 

32.9 



93 

94.6 
334 
371 
14-3 
4475 
68.8 
92.2 
31-2 

23-3 



122 

94-5 
334 
401 

14.0 

4655 
no. 8 
148.5 

27.9 

20.8 



46.5 

97.1 
336 

39i 
14. 1 
4625 
123 
164.9 
27.2 

20.3 



CHAPTER XVI. 
FRICTION OF ENGINES. 

THE efficiency and economy of steam-engines are com- 
monly based on the indicated horse-power, because that 
power is a definite quantity that may be readily determined. 
On the other hand, it is usually difficult and sometimes im- 
possible to make a satisfactory determination of the power 
actually delivered by the engine. A common way of deter- 
mining the work consumed by friction in the engine itself is 
to disconnect the driving-belt, or other gear for transmitting 
power from the engine, and to place a friction-brake on the 
main shaft; the power developed is then determined by aid 
of indicators, and the power delivered is measured by the 
brake, the difference being the power consumed by friction. 
Such a determination for a large engine involves much 
trouble and expense, and may be unsatisfactory since the 
engine-friction may depend largely on the gear for transmit- 
ting power from the engine, especially when belts or ropes are 
used for that purpose. 

The friction of a pumping-engine may be determined from 
a comparison of the indicated power of the steam-cylinders 
with the indicated work of the pumps, or, better, with the 
work done in lifting water from the well and delivering it to 
the forcing-main. But the friction thus determined is the 
friction of both the engine and the pump. Air-compressors 
and refrigerating machines may be treated in the same way 
to determine the friction of both engine and compressor. 
Again, the combined friction of an engine and a directly 
connected electric generator may be determined by compar- 
ing the indicated power of the engine with the electric 

429 



430 



THERMODYNAMICS OF THE STEAM-ENGINE. 



output of the generator, allowing for electricity consumed or 
wasted in the generator itself. 

The friction of a steam-engine may consume from 5 to 15 
per cent of the indicated horse-power, depending on the type 
and condition of the engine. The power required to drive 
the air-pump (when connected to the engine) is commonly 
charged to the friction of the engine. It is usual to consider 
that seven per cent of the indicated power of the engine is 
expended on the air-pump. Independent air-pumps which 
can be driven at the best speed consume much less power; 
those of some United States naval vessels used only one or 
two per cent of the power of the main engines. But as inde- 
pendent air-pumps are usually direct-acting steam-pumps, 
much of the apparent advantage just pointed out is lost on 
account of the excessive steam-consumption of such pumps. 

Mechanical Efficiency. — The ratio of the power delivered 
by an engine to the power generated in the cylinder is the 
mechanical efficiency; or it may be taken as the ratio of the 
brake to the indicated power. The mechanical efficiency of 
engines varies from 0.85 to 0.95, corresponding to the per 
cent of friction given above. 

The following table gives the mechanical efficiencies of a 

Table XLV. 

MECHANICAL EFFICIENCIES OF ENGINES. 



Kind of engine. 



Simple engines: 

Horizontal portable 

Hoadley 

High-speed, straight-Hne 

Corliss condensing 

" non-condensing 

Compound: 

Portable 

Semi-portable 

Horizontal 

Horizontal mill-engine 

Schmidt, superheated steam 

Leavitt pumping-engine 

Triple-expansion Leavitt pumping-engine 



Horse- power. 


Efficiency. 


24 


O.86 


80 


O.91 


56 


O.96 


160 


0.8l 


IOO 


0.86 


78 


0.88 


60 


0.88 


59 


0.90 


288 


0.86 


no 


0.92 


643 


o-93 


576 


0.90 



FRICTION OF ENGINES. 43 l 

number of engines, determined by brake tests, or, in case of 
the pumping-engines, by measuring the work done in pump- 
ing waters. 

Initial Friction and Load Friction. — A part of the fric- 
tion of an engine, such as the friction of the piston-rings and' 
at the stuffing-boxes of piston-rods and valve-rods, may be 
expected to remain constant for all powers. The friction at 
the cross-head guides and crank-pins is due mainly to the 
thrust or pull of the steam-pressure, and will be nearly pro- 
portional to the mean effective pressure. Friction at other 
places, such as the main bearings, will be due in part to weight 
and in part to steam-pressure. On the whole it appears 
probable that the friction may be divided into two parts, of 
which one is independent of the load on the engine, and the 
other is proportional to the load. The first may be called the 
initial friction and the second, the load friction. Progressive 
brake tests at increasing loads confirm this conclusion. 

Table XLVI gives the results of tests made by Walther- 
Meunier and Ludwig* to determine the friction of a horizon- 
tal-receiver compound engine, with cranks at right angles and 
with a fly-wheel, grooved for rope-driving, between the 
cranks. The piston-rod of each piston extended through the 
cylinder-cover and was carried by a cross-head on guides, and 
the air-pump was worked from the high-pressure piston-rod. 
The cylinders each had four plain slide-valves, two for admis- 
sion and two for exhaust; the exhaust-valves had a fixed 
motion, but the admission-valves were moved by a cam so 
that the cut-off was determined by the governor. 

The main dimensions of the engine were: 

Stroke i.i metres. 

Diameter : small piston 0.536 

large piston 0.800 

piston-rods 0.080 

Diameter, air-pump pistons 0.360 

Stroke, air-pump 0.476 

Diameter, fly-wheel 6.610 

* Bulletin de la Soc. Ind. de Mulkouse, vol. lvii, p. 140. 



432 THERMODYNAMICS OF THE STEAM-ENGINE. 

Table XLVI. 

FRICTION OF COMPOUND ENGINE. 

Walther-Meunier and Ludwig, Bulletin de la Soc. Ind. de Mulhouse> 

vol. lvii, p. 140. 





Condition. 


Horse-powers — Chevaux aux vapeur. 


Friction. 






Indicated. 


Effective. 


Absorbed 
by engine. 


Efficiency. 


I 

2 

3 
4 
5 
6 

7 
8 

9 


Compound 
condensing 

with 
air-pump. 


288.5 
276.9 
265.6 

243-7 
222.7 
201.5 
180.4 
158.1 
136. 1 


249.O 

238.9 
228.9 
208.8 
188.7 
168.6 
148.5 
128.4 
IO8.3 


39 
38 
36 
34 
34 
32 
3i 
29 
27 


5 


7 

9 


9 

9 

7 
8 


O.137 
O.138 
O.I39 
O.144 

O.I53 
O.164 
O.178 
O.189 
O.205 


O.863 
O.862 
O.861 
O.856 

O.847 
O.836 
O 822 
O.811 
0.795 


10 
n 
12 
13 
14 
15 
16 

17 

18 

19 


High- 

pressure 

cylinder 

only. 

Condensing 

with 
air-pump. 


I53-I 
142.0 
130.9 
120. 1 
109,0 

97-5 
86.3 

75-7 
65-5 
55-2 


128.4 
Il8. 3 
IO8.3 
98.2 
88.2 
78.1 
68.1 
58.0 
48.0 
37-9 


24 

23 
22 
21 
20 

19 

18 

17 
17 
17 


7 
7 
6 

9 

8 

4 

3 
7 
5 
3 


O.161 
O.167 
O.173 
O.182 

0. 191 

O.I99 
0.2I2 
O.234 
O.267 

0.3I3 


O.839 

O.833 
O.827 
O.818 
O.809 
O.801 
O.788 
O.766 

0.733 
O.687 


20 
21 
22 

23 
24 

25 
26 
27 
28 
29 


High- 
pressure 
cylinder 
only. 
Non- 
condensing, 
no air-pump. 


145-9 

135-7 

125 2 

114.4 

103.9 

93-o 

82.0 

71.7 

61.6 

51.3 


128.4 
118. 3 
108.3 
98.2 
88.2 
78.1 
68.1 
58.0 
48.0 
37-9 


17 
17 
16 
16 
15 
14 
13 
13 
13 
13 


5 
4 
9 
2 

7 
9 
9 
7 
6 

4 


0.I20 
0. 129 

0.135 
O.I42 
O.I52 
O.160 
O.I70 
0. 191 
0.22I 
O.262 


O.880 
O.871 
O.865 
O.858 
O.848 
O.840 
O.83O 
O.809 
O.779 
0.738 



The engine during the experiments made 58 revolutions 
per minute. The air-pump had two single-acting vertical 
pistons. 

Each experiment lasted 10 or 20 minutes, during which 
the load on the brake was maintained constant, and indicator- 
diagrams were taken. The experiments with small load on 



FRICTION OF ENGINES. 



433 



the brake (numbers 9, 18, 19, 28, and 29) were difficult and 
uncertain. 

The first nine tests were made with the engine working 
compound. Tests 10 to 19 were made with the high-pressure 
cylinder only in action and with condensation, the low-pres- 
sure connecting-rod being disconnected. Tests 20 to 29 were 
made with the high-pressure cylinder in action, without con- 
densation. 

The results of these tests are plotted on Fig. 84, using the 




ABSCISSAE, EFFECTIVE HORSEPOWER. 
ORDINATES, FRICTION HORSEPOWER. 



50 

l 



100 

I 



150 

1 



200 



250 



Fig. 84. 

effective horse-power for abscissae and the brake horse-powers 
for ordinates. Omitting tests with small powers (for which 
the brake ran unsteadily), it appears that each series of tests 
can be represented by a straight line which crosses the axis of 
ordinates above the origin; thus affording a confirmation of 
the assumption that an engine has a constant initial friction, 
and a load friction which is proportional to the load. 

Now the initial friction which depends on the size and 
construction of the engine may be assumed to be propor- 



434 THERMODYNAMICS OF THE STEAM-ENGINE. 

tional to the normal net or brake horse-power, P n , which the 
engine is designed to deliver, and may be represented by 

aP n , 

where a .is a constant to be determined from a diagram like 
Fig. 84. If P is the net horse-power delivered by the engine 
at any time, then the load friction corresponding is 

where b is a second constant to be determined from experi- 
ments. The total friction of the engine will be 

F=aP n +bP, (308) 

so that the indicated power of the engine will be 

I.H.P. = P+aP n +bP=aP n +(i + b)P. (309) 

The mechanical efficiency corresponding will be 

I.H.P. - F _ P 
Vm ~~ " I.H.P. " ~~ I.H.P.' ' ' * ^ 3I °) 

The compound condensing engine for which tests are 
represented by Fig. 84 developed 290 I.H.P. and delivered 
250 horse-power to the brake, so that 40 hnrse-power was 
consumed in friction. The diagram shows also that the initial 
friction was 20 horse-power, and consequently the load fric- 
tion was 20 horse-power. The values of a and b are conse- 
quently 

a == 20^250 = 0.07 ; 

b = (40 — 20) -=- 250 = 0/07. 
The indicated horse-power for a given load Pis 

I.H.P. = o.oyP H + 1.07P. . . . (311) 

Similar equations can be deduced for the engine with 
steam supplied to the small cylinder only; but as the engine 
is not then in normal condition they are not very useful. 

The maximum efficiency of this engine is 

250 -r- 290 = 0.86; 



FRICTION OF ENGINES. 



435 



but at half load (125 horse-power) the indicated horse-power is 

I.H.P. = 0.07 X 250+ 1.07 X 125 = 151, 

and the efficiency is 

125 -r- 151 = 0.83. 

Table XL VII. 

FRICTION OF CORLISS ENGINE AT CREUSOT. 

By F. DELAFOND, Annates des Mines, ii 

Condensing with air-pump, tests 1-33. 
Non-condensing without air-pump, tests 34-46. 











Horse-power — Cheval 2 


vapeur. 




Cut-off frac- 
tion of 
stroke. 


Pressure at 

cut-off. kilos 

per sq. cm. 


Revolutions 
per minute. 










Indicated. 


Effective. 


Absorbed 
by engine. 














I 


0.039 


0.64 


64.0 


27.8 


16.3 


"•5 


2 


0.044 


2.40 


68.5 


60.0 


37-6 


22.4 


3 


0.044 


2.90 


65.0 


67.2 


45-2 


22.0 


4 


0.065 


4.90 


64.0 


117. 


88.7 


28.3 


5 


0.065 


6.20 


61.0 


138.5 


106.3 


32.2 


6 


0.065 


7. 10 


64.0 


163.2 


129.2 


34'0 


7 


0.065 


7.60 


64.0 


185.0 


144.6 


40.4 


8 


O.IOO 


0. 16 


58.0 


21.0 


10.6 


10.4 


9 


0.106 


i-55 


60.0 


61.9 


42.3 


19.6 


10 


O.IOO 


2.82 


57-3 


82.7 


61.0 


21.7 


11 


0.096 


4.80 


58.3 


x 35«3 


106.7 


28.6 


12 


0. 128 


4.82 


58.3 


154-5 


124.8 


29.7 


13 


0. 142 


0.76 


62.0 


42.3 


28.4 


i3»9 


14 


0.137 


0.71 


60.6 


44-3 


28.7 


15.6 


*5 


0.132 


2.50 


54-o 


79-5 


59-8 


19.7 


16 


0.147 


2.60 


61.6 


100. 


78.2 


21.8 


17 


0.155 


4-65 


60.0 


177.2 


145.0 


32.2 


18 


0.167 


0.22 


61.0 


40.2 


27.9 


12.3 


19 


0.197 


2-55 


57-2 


no. 8 


83-3 


27-5 


20 


0.273 


0.40 


62.3 


50.2 


33-8 


16.4 


21 


0.264 


i-57 


63-3 


89.1 


6r.8 


27-3 


22 


0.240 


1 .64 


62.0 


87.2 


63.1 


24.1 


23 


0.245 


3- 2 5 


56.0 


145.0 


116. 


29.0 


24 


0.260 


4.76 


58.0 


209.4 


178.0 


3i-4 


■25 


°-335 


0.25 


59.0 


47.2 


32.5 


14.7 


26 


o.339 


1.94 


58.3 


in. 7 


90.0 


21.7 


27 


0.338 


2.97 


61.0 


161. 8 


133.0 


28.8 


28 


1 


0.47 


59-3 


81.3 


67.2 


14. 1 


29 


1 


0.47 


61.0 


80.8 


67.9 


12.9 


30 


1 


1.60 


61.6 


148.5 


128.4 


20.1 


3 1 


1 


2.70 


61.5 


216.5 


191.0 


25-5 


32 


1 


2.70 


61.5 


2155 


191. 


24-5 


33 


0.50 


0.70 


61.5 


15-8 


0.0 


15.8 


34 


0.120 


6.00 


60.0 


132.5 


107.5 


25.0 


35 


0.106 


7.00 


53-0 


125.0 


103.0 


22.0 


36 


0. 120 


7-5o 


62.0 


172.0 


148.0 


24.0 


37 


0.150 


4-57 


55-o 


102.3 


86.5 


15.8 


38 


0.262 


4-5o 


59 


149.2 


132.3 


16.9 


39 


0.293 


4 55 


59-° 


171. 8 


153-8 


18.0 


40 


Q-37 1 


4.40 


60.0 


195-3 


177.2 


18. 1 


4i 


0.348 


2-75 


58.0 


85.1 


73-i 


12.0 


42 


0.348 


2-75 


58.5 


84.8 


71. 1 


*3-7 


43 


0.440 


3-48 


62.0 


151. 


134.3 


,16.7 


44 


0. in 


3-30 


62.0 


12.8 


0.0 


12.3 


45 


0.50 


1.20 


62.0 


12.3 


0.0 


12.3 


46 


1 


0.50 


62.0 


10.45 


0.0 


10.45 



43 6 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table XLVII gives the results of a large number of brake 
tests made on a Corliss engine at Creusot by M. F. Delafond, 
both with and without a vacuum, and with varying steam- 
pressures and cut-off. The tests with a vacuum are plotted 
on Fig. 85, and those without a vacuum are given in Fig. 86. 
In both figures the abscissae are the indicated horse-powers, and 
the ordinates are the friction horse-powers. Most of the tests 
are represented by dots; those tests which were made with 
the most economical cut-off (one sixth for the engine with 

40 



35 



30 



25 



20 



15 



10 







































• 
















• 
















• 


+ * 


• 


•^^ 1 


• 


® 










• 










® 






1 


• 


• ^s 


■• • 
















• - 








w 










+y 






Absci 


ssae, in 


dicatec 


. horsei 


jower 




• 




c 
( 


•> 


Ordin 


ates, fr 


iction 1 


lorsepo 


wer 











































20 



40 



60 



80 100 
FiG. 85. 



120 



140 



160 



180 



200 



condensation and one third without) are represented by 
crosses. A few tests with very long cut-off, on Fig. 84, are 
represented by circles. The straight lines on both figures are 
drawn to represent the tests indicated by crosses. In general 
the points representing tests with short cut-off and high 
steam-pressure lie above the lines, and points representing 
tests with long cut-off and low steam-pressure lie below the 
lines, though there are some notable exceptions to this rule. 
The circles on Fig. 84, representing tests with cut-off near the 



FRICTION OF ENGINES. 



437 



end of the stroke, show much less friction than the other tests. 
The tests on this engine show clearly that both initial and 
load friction are affected by the cut-off and the steam-pres- 
sure, and that friction tests should be made at the cut-off 
which the engine is expected to have in service. 

35 



20 



15 



10 



















• 
















• 


















• 








+ 










+ 




















+ 




















Al 


>scissat 


:, indie. 


ited ho 


rsepow 


IT 










Or 


linates 


, fricti< 


m hors 


ipower 

























20 



40 



60 



80 100 
Fig. 86. 



120 



140 



160 



180 



200 



The initial friction was eight horse-power both with and 
without condensation. But Table XXX shows that the 
engine with condensation gave the best economy when it 
indicated 160 horse-power; the friction was then 30 horse- 
power, so that the net horse-power was 130, which will be 
taken for the normal horse-power P n . Consequently 



a — 8 -r- 130 = 0.06; 

b = (30 - 8) -^ 130 = 0.17. 

. I.H.P. = o.o62F n + 1.17P. 



(312) 



In like manner Table XLVII shows the best economy 
without condensation, for about 100 indicated horse-power, 
for which the friction is 14 horse-power, leaving 86 for the 
normal power of the engine. Consequently 

a — 8 -r- 86 = 0.09; 

b — (14 — 8) H- 86 = 0.07. 

.-. I.H.P. = 0.09^+ 1. 07P. . o . (313) 



438 



THERMODYNAMICS OF THE STEAM-ENGINE. 



The increased value of a for the non-condensing engine is 
due directly to the reduction in the economical power of the 
engine; on the other hand, the reduction in the value of b is. 
due to the omission of the air-pump. 

Thurston's Experiments. — As a result of a large number 
of tests on non-condensing engines, made under his direction 
or with his advice, Prof. R. H. Thurston* concludes that, 
for engines of that type, the friction is independent of the 
load, and that it can, in practice, be determined by indicating 
the engine without a load. 



Table XLVIII. 

FRICTION OF NON-CONDENSING ENGINE. 

STRAIGHT-LINE ENGINE, 8 INCHES DIAMETER, 14 INCHES STROKE. 



No. of 


Boiler- 


Revolutions. 


Brake H. P. 


I. H. P. 


Frictional H. P. 


Diagram. 


pressure. 










I 


50 


332 


4.06 


7.41 


3-35 


2 


65 


229 


4.98 


7.58 


2.60 


3 


63 


230 


6.00 


IO.OO 


4.00 


4 


69 


230 


7.00 


IO.27 


3-27 


5 


73 


230 


8.IO 


n-75 


3-65 


6 


77 


230 


9.00 


12.70 


3-7o 


7 


75 


230 


IO.OO 


14.02 


' 4.02 


8 


80 


230 


II.OO 


14.78 


3-78 


9 


80 


230 


I2.00 


15-17 


3-17 


10 


85 


230 


13.OO 


15.96 


2.96 


n 


75 


230 


14.OO 


16.86 


2.86 


12 


70 


230 


15.OO 


17.80 


2.80 


13 


72 


231 


20. IO 


22.07 


1.97 


14 


75 


230 


25.OO 


28.31 


3-3i 


15 


60 


229 


29-55 


33-04 


3-40 


16 


58 


229 


34-86 


37.20 


2-34 


17 


70 


229 


39-85 


43-04 


3-19 


18 


85 


230 


45.00 


47-79 


2.78 


19 


90 


230 


50.00 


52.60 


2.60 


20 


85 


230 


55- 00 


57-54 


2-54 



Table XLVIII gives the details of one series of tests. 
The friction horse-power is small in all the tests, and the 
variations are small and irregular, and appear to depend on 
the state of lubrication and other minor causes rather than on 
the change of load. 

* Trans, of the Am. Soc. of Meek. Engrs., vols, viii, ix, and x. 



FRICTION OF ENGINES. 439 

Distribution of Friction. — As a consequence of his con- 
clusion in the preceding section, Professor Thurston says 
that the friction of an engine may be found by driving it from 
some external source of power, with the engine in substan- 
tially the same condition as when running as usual, but with- 
out steam in its cylinder, and by measuring the power 
required to drive it by aid of a transmission dynamometer. 
Extending the principle, the distribution of friction among the 
several members of the engine may be found by disconnecting 
the several members, one after another, and measuring the 
power required to run the remaining members. 

The summary of a number of tests of this sort, made by 
Prof. R. C. Carpenter and Mr. G. B. Preston, are given in 
Table XLIX. Preliminary tests under normal conditions 
showed that the friction of the several engines was practically 
the same at all loads and speeds. 

The most remarkable feature in this table is the friction of 
the main bearings, which in all cases is large, both relatively 
and absolutely. The coefficient of friction for the main 
bearings, calculated by the formula 

33000 H.P. 
* ~ ~ pen ' 

is given in Table L. / is the pressure on the bearings in 
pounds for the engines light, and//^ the mean pressure on 
the piston for the engines loaded; c is the circumference of 
the bearings in feet; n is the number of revolutions per 
minute; and H.P. is the horse-power required to overcome 
the friction of the bearings. 

The large amount of work absorbed by the main bearings 
and the large coefficient of friction appear the more remark- 
able from the fact that the coefficient of friction for car-axle 
journals is often as low as one-tenth of one per cent, the 
difference being probably due to the difference in the methods 
of lubrication. 



44Q 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Table XLIX. 

DISTRIBUTION OF FRICTION. 



Parts of Engine. 



Main Bearings 

Piston and Rod 

Crank Pin 

Cross Head and Wrist Pin 

Valve and Rod 

Eccentric Strap 

Link and Eccentric 

Air Pump , 

Total 



Percentages of Total Friction. 





^ 


= A i: 


c o 


c u 


X > 

« "3 
or 


« (J 


Lansing Ire 
-Traction- L 
e Valve-gea 


Lnnsing Iro 
— Automat 
ed V;tlve. 


Lansing Iro 
- Condensin 
d Valve. 


.3*3 


1 > 


; i»o 


; ' <U 


S>g 


-b-XZ 


00-^ ~ 

* >~ a 


en 




rt c 

C/2 


" x 1 
Wor 
com 


x °~ 


i"x 

Wor 
Bala 






t^ 


" 


<N 


47.0 


35-4 


35-o 


41 .6 


46.0 


32.9 


25.0 


21.0 


49.1 




6.8 


5-i 


13.0 


21.8 


5-4 


4.1 






2-5 

5-3 


26.4 
4.0 








22.0 


9-3 


21.0 






9.0 














I2.0 


100. 


100. 


100. 


100.0 


IOO. O 



Table L. 

COEFFICIENT OF FRICTION FOR THE MAIN BEARINGS OF 

STEAM-ENGINES. 











a 

§ 


en 
13 

c 


*c3 

c 

1* 




■ *a" 

'CO 


1 

u 

3 . 
O it 












i~ 


3 


&-"T 


&H 5 


►— i*J 










„• 


3 VJ 




_i-J . 





v« 3 






Engine. 




4) rt 
3 G 


OT3 
X! C 

b/o— ' 


S3 

u 

S.S 

g 
c5 


<u 

JTSo 
.12 a 


c a 
c 


O 3 
O u 

3 a 










ad 


'5 


Ss 


c 


m 

> 3 






• 




fe 


£ 


Q 


U-2 


Uo 


ot 


fV 


XI2" 
XI8" 






0.85 
3-70 


1500 
2600 


3 

5 


IO 


06 


230 

190 


*I2' 


Automatic (L. I 


. W.) .. 


.19 


•05 


7' 


X 10" 


Traction (L. I. 


W.).... 


0.68 


500 


2f 


•31 


.08 


200 


21' 


X 20" 


Condensing (L. 


I. W.) 


3-30 


4000 


5* 


.09 


. -04 


206 



*The 12" X 18" automatic engine was new, and gave, throughout, an ex- 
cessive amount of friction as compared with the older engines of the same class 
and make. 



FRICTION OF ENGINES. 44 T 

The second and obvious conclusion from Table L is that 
the valve should be balanced, and that nine tenths of the 
friction of an unbalanced slide-valve is unnecessary waste. 

The friction of the piston and piston-rod is always con- 
siderable, but it varies much with the type of the engine, 
and with differences in handling. It is quite possible to 
change the effective power of an engine by screwing up the 
piston-rod stuffing-box too tightly. The packing of both 
piston and rod should be no tighter than is necessary to pre- 
vent perceptible leakage, and is more likely to be too tight 
than too loose. 



CHAPTER XVII. 
COMPRESSED AIR. 

COMPRESSED air is used for transmitting power, for stor- 
ing energy, and for producing refrigeration. Air at moderate 
pressure, produced by blowing-engines, is used in the produc- 
tion of iron and steel; and currents of air at slightly higher 
pressure than that of the atmosphere (produced by centrifugal 
fan-blowers) are used to ventilate mines, buildings, and 
ships, and for producing forced draught for steam-boilers. 
Attention will be given mainly to the transmission and storage 
of energy. The production and use of ventilating currents 
require and are susceptible of but little theoretical treatment. 
Refrigeration will be reserved for another chapter. 

A treatment of the transmission of power by compressed 
air involves the discussion of air-compressors, of the flow of 
air through pipes, and of compressed-air engines or motors. 
The storage of energy differs from the transmission of power 
in that the compressed air, which is forced into a reservoir at 
high pressure, is used at a much lower pressure at the air- 
motor. 

Air-compressors. — There are three types of machines 
used for compressing or moving air: (i) piston air-compressors, 
(2) rotary blowers, (3) centrifugal blowers or fans. 

The piston air-compressor is always used for producing 
high pressures. It consists of a piston moving in a cylinder 
with inlet- and exit-valves at each end. Commonly the valves 
are actuated by the air itself, but some compressors have their 
valves moved mechanically. Blowing-engines are usually 

442 



COMPRESSED AIR, 443 

piston-compressors, though the pressures produced are only- 
ten or twenty pounds per square inch. 

Rotary blowers have one or more rotating parts, so 
arranged that as they rotate, chambers of varying capacity 
are formed which receive air at atmospheric pressure, com- 
press it, and deliver it against a higher pressure. They are 
simple and compact, but are wasteful of power on account of 
friction and leakage, and are used only for moderate pressures. 

Fan-blowers consist of a number of radial plates or vanes 
fixed to a horizontal axis and enclosed in a case. When the 
axis and the vanes attached to it are rotated at a high speed 
air is drawn in through openings near the axis and is driven 
by centrifugal force into the case, from which it flows into the 
delivery-main or duct. Only low pressures, suitable for 
ventilation and forced draught, can be produced in this way. 
But little has been done in the development of the theory or 
the determination of the practical efficiency of fan-blowers. 
Some ventilating-fans have their axes parallel to the direction 
of the air-current, and the vanes have a more or less helicoidal 
form, so that they may force the air by direct pressure; they 
are in effect the converse of a windmill, producing instead of 
being driven by the current of air. They are useful rather 
for moving air than for producing a pressure. 

Fluid Piston-compressors. — It will be shown that the 
effect of clearance is to diminish the capacity of the com- 
pressor; consequently the clearance should be made as small as 
possible. With this in view the valves of compressors and 
blowers are commonly set in the cylinder-heads. Single- 
acting compressors with vertical cylinders have been made with 
a layer of water or some other fluid on top of the piston, which 
entirely fills the clearance-space when the piston is at the end 
of the stroke. An extension of this principle gives what are 
known as fluid piston-compressors. Such a compressor com- 
monly has a double-acting piston in a horizontal cylinder 
much longer than the stroke of the piston, thus giving a large 
clearance at each end. The clearance-spaces extend upward 



444 THERMODYNAMICS OF THE STEAM-ENGINE. 

to a considerable height, and the admission- and exhaust- 
valves are placed at or near the top, and the entire clearance- 
space is filled with water. The spaces and heights must be 
so arranged that when the piston is at one end of its stroke 
the water at that end shall fill the clearance and cover the 
valves, and at the other end the water shall not fall to the 
level of the top of the cylinder. There are consequently two 
vertical fluid pistons actuated by a double-acting horizontal 
piston. It is essential that the spaces in which the fluid 
pistons act shall give no places in which air may be caught as 
in a pocket, and that there are no projecting ribs or other 
irregularities to break the surface of the water; and, further, 
the compressor must be run at a moderate speed. The water 
forming the fluid pistons becomes heated and saturated with 
air by continuous use, and should be renewed. 

Air-pumps used with condensing-engines or for other 
purposes may be made with fluid pistons which are renewed 
by the water coming with the air or vapor. In case the water 
thus supplied is insufficient, water from without may be 
admitted, or water from the delivery may be allowed to flow 
back to the admission side of the pump. 

Displacement-compressors. — When a supply of water 
under sufficient head is available, air may be compressed in 
suitably arranged cylinders or compressors by direct action of 
the water on air, compressing it and expelling it by displace- 
ment. Such compressors are very wasteful of power, and in 
general it is better to use water-power for driving piston- 
compressors, properly geared to turbine-wheels or other 
motors. 

Cooling during Compression. — There is always a con- 
siderable rise of temperature due to compressing air in a 
piston air-compressor, which is liable to give trouble by heat- 
ing the cylinder and interfering with lubrication. Blowing- 
engines which produce only moderate pressures usually have 
their cylinders lubricated with graphite, and no attempt is 
made to cool them. All compressors which produce high 



COMPRESSED AIR. 445 

pressures have their cylinders cooled either by a water-jacket 
or by injecting water, or by both methods. 

Since the air after compression is cooled either purposely 
or unavoidably, there would be a great advantage in cooling 
the air during compression, and thereby reducing the work of 
compression. Attempts have been made to cool the air by 
spraying water into the cylinder, but experience has shown 
that the work of compression is not much affected by so 
doing. The only effective way of reducing the work of com- 
pression is to use a compound compressor, and to cool the air 
on the way from the first to the second cylinder. Three- 
stage compressors are used for very high pressures. It is, 
however, found that air which has been compressed to a high 
pressure and great density is more readily cooled during com- 
pression. 

Moisture in the Cylinder. — If water is not injected into 
the cylinder of an air-compressor the moisture in the air will 
depend on the hygroscopic condition of the atmosphere. 
But even if the air were saturated with moisture the absolute 
and the relative weight of water in the cylinder would be 
insignificant. Thus at 6o° F. the pressure of saturated steam 
is about one-fourth of a pound per square inch, and the weight 
of one cubic foot is about 0.0008 of a pound, while the weight 
of one cubic foot of air is about 0.08 of a pound. If the 
atmosphere is not saturated, then the watery vapor drawn into 
the compressor with the air will follow the laws of superheated 
steam. Now the adiabatic equations for air and for super- 
heated steam are 

^1.405 _ ^^1.405 anc j prf—p^^ 

so that the only effect of moisture in the air will be to slightly 
reduce the exponent of the adiabatic equation. This conclu- 
sion probably holds when the cylinder is cooled by a water- 
jacket. 

When water is sprayed into the cylinder of a compressor 
the temperature of the air and the amount of vapor mixed 



446 THERMODYNAMICS OF THE STEAM-ENGINE. 

with it vary, and there is no ready way of determining its 
condition. But, as has been stated, the spraying of water 
into the cylinder does not much reduce the work of compres- 
sion, and consequently it is probable we can assume that the 
compression always follows the law expressed by an exponen- 
tial equation ; such as 

pv n = p 1 v 1 H . 

The value to be given to n is not well known; it may be 
as small as 1.2 for a fluid piston-compressor, and it may 
approach 1.4 when the cooling of the air is ineffective, as is 
usually the case. 

Power Expended. — The indicator-diagram of an air- 
compressor with no clearance-space is represented by Fig. 87. 
Air is drawn in at atmospheric pressure 
in the part of the cycle of operations repre- 
sented by dc\ in the part represented by cb 
the air is compressed, and in the part repre- 
Fig. 87. sented by ba it is expelled against the 

higher pressure. 

If p x is the specific pressure and v x the specific volume of 
one pound of air at atmospheric pressure, and /, and v 9 
corresponding quantities at the higher pressure, then the work 
done by the atmosphere on the piston of the compressor 
while air is drawn in is p x v x . Assuming that the compression 
curve cb may be represented by an exponential curve having 

the form 

pv n = p x v x = const., 

then the work of compression is 




A v * i (P 



n—\ \\p x 
The work of expulsion from b to a is 

I M — T 

'A\n .. ffi 



'■-'} 



'•'•= "ijf ='■•€)" 



COMPRESSED AIR. 



447 



The effective work of the cycle is therefore 



W 



Pf>x { IP 



n — I j \p x 



\ +*>$}"'' -**.-. 



W = p x v x 



n 



n — i ( \p, 



& 



— i 



(314) 



Equation (314) gives the work done to compress one 
pound of air, p x and /„ being specific pressures (in pounds per 
square foot), and v x the specific volume, which may be 
calculated by aid of the equation 



pv 
T 



T 



in which the subscripts refer to the normal properties of air 
at freezing-point and at atmospheric pressure. 

If, instead of the specific volume v lf we use the volume 
V x of air drawn into the compressor we may readily transform 
equation (314) to give the horse-power directly, obtaining 



H. P. = 



I44A V x n 
33000( 



n^T)\[Jj - 1 }' ' ' (315 ) 



where p x is the pressure of the atmosphere in pounds per 
square inch, and n is the exponent of the equation represent- 
ing the compression curve, which may vary from 1.4 for dry- 
air compressors to 1.2 for fluid piston-compressors. 

Effect of Clearance. — The indicator-diagram of an air- 
compressor with clearance may be represented by Fig. 88. 
The end of the stroke expelling air is at a, 
and the air remaining in the cylinder ex- 
pands from a to d, till the pressure becomes 
, equal to the pressure of the atmosphere 
before the next supply of air is drawn in. 
The expansion curve ad may commonly be 
represented by an exponential equation having the same 
exponent as the compression curve cb, in which case the air 




Fig. 



448 THERMODYNAMICS OF THE STEAM-ENGINE. 

in the clearance acts as a cushion which stores and restores 
energy, but does not affect the work done on the air passing 
through the cylinder. The work of compressing one unit of 
weight of air in such a compressor may be calculated by aid 
of equation (312), but the equation (315) for the horse-power 
cannot be used directly. 

The principal effect of clearance is to increase the size of 
the cylinder required for a certain duty in the ratio of the 
entire length of the diagram in Fig. 88 to the length of the 
line dc. 

Let the clearance be — part of the piston displacement. 

At the beginning of the filling stroke, represented by the 
point a> the volume will be filled with air at the pressure / 2 . 
After the expansion represented by ad the air in the clearance 
will have the pressure p x , and, assuming that the expansion 
follows the law expressed by the exponential equation 

its volume will be 

L (hV 

m\pj 

part of the piston displacement. The ratio of the line dc to 
the length of the diagram will consequently be 

* =I _i(£4 + ± ; (3l6 ) 

ac m \p j ' m w / 

and this is the factor by which the piston displacement cal- 
culated without clearance must be divided to find the actual 
piston displacement. 

Temperature at the End of Compression. — When the 
air in the compressor-cylinder is dry or contains only the 
moisture brought in with it, it may be assumed that the 
mixture of air and vapor follows the law of perfect gases, 

pv_p J v 1 



COMPRESSED AIR. 449 

which, combined with the exponential equation 

pv n — p x v, n , 
gives 



T >= T >(j) (317) 

from which the final temperature T 9 at the end of compression 
may be determined when 7", is known. When water is used 
freely in the cylinder of a compressor the final temperature 
cannot be determined by calculation, but must be determined 
from tests on compressors. 

Contraction after Compression. — Ordinarily compressed 
air loses both pressure and temperature on the way from the 
compressor to the place where it is to be used. The loss of 
pressure will be discussed under the head of the flow of air in 
long pipes; it should not be large, unless the air is carried 
a long distance. The loss of temperature causes a contraction 
of volume in two ways: first, the volume of the air at a given 
pressure is inversely as the absolute temperature; second, the 
moisture in the air (whether brought in by the air or supplied 
in the condenser) in excess of that which will saturate the air 
at the lowest temperature in the conduit, is condensed. 
Provision must be made for draining off the condensed water. 
The method of estimating the contraction of volume due to 
the condensation of moisture will be exhibited later in the 
calculation of a special problem. 

Interchange of Heat. — The interchanges of heat between 
the air in the cylinder of an air-compressor and the walls of 
the cylinder are the converse of those taking place between 
the steam and the walls of the cylinder of a steam-engine, and 
are much less in amount. The walls of the cylinder are never 
so cool as the incoming air nor so warm as the air expelled; 
consequently the air receives heat during admission and the 
beginning of compression, and yields heat during the latter 
part of compression and during expulsion. The presence of 
moisture in the air increases this effect. 



450 THERMODYNAMICS OF THE STEAM-ENGINE. 

Volume of the Compressor-cylinder. — Let a compressor 
making n revolutions per minute be required to deliver V 3 
cubic feet of air at the temperature t° F., or T t ° absolute, 
and at the absolute pressure^ pounds per square inch, at the 
place where the air is to be used. Assuming that the air is 
dry when it is delivered and that the atmosphere is dry when 
it is taken into the compressor, then the volume drawn into 
the compressor per minute at the temperature T 1 and the 
pressure p x will be 

f ' =f #: (3i8) 

cubic feet; and this expression will be correct whatever may 
be the intermediate temperatures, pressures, or condition of 
saturation of the air. 

If the compressor has no clearance the piston displacement 
will be 

V 

^ (319) 

if the clearance is — part of the piston displacement, dividing 

m r ° 

by the factor (316) gives for the piston displacement 

Z + lt-Lfift + L] (320) 

2n I m \pj ' w) w } 

expressed in cubic feet. 

The pressure in the compressor-cylinder when air is drawn 
in is always less than the pressure of the atmosphere, and 
when the air is expelled it is greater than the pressure against 
which it is delivered. From these causes and from other 
imperfections the compressor will not deliver the quantity of 
air calculated from its dimensions, and consequently the 
volume of the cylinder as calculated, whether with or without 
clearance, must be increased by an amount to be determined 
by experiment. 



COMPRESSED AIR. 45 I 

Compound Compressors. — When air is to be compressed 
from the pressure p x to the pressure p„ but is to be delivered 
at the initial temperature t„ the work of compression may be 
reduced by dividing it between two cylinders, one of which 
takes the air at atmospheric pressure and delivers it at an 
intermediate pressure p' to a reservoir, from which the other 
cylinder takes it and delivers it at the required pressure^, 
provided that the air be cooled, at the pressure p' , between 
the two cylinders. 

The proper method of dividing the pressures and of pro- 
portioning the volumes of the cylinders so that the work of 
compression may be reduced to a minimum may be deduced 
from equation (314) when there is no clearance or when the 
clearance is neglected. 

The work of compressing one pound of air from the pres 
sure/j to the pressure/' is 

The work of compressing one pound from the pressure p' 
to /, is 

because the air after compression in the first cylinder is cooled 
to the temperature /, before it is supplied to the second 
cylinder, and consequently p'v' = p x v x . The total work of 
compression is 

w = Wx + w % =a % -^ I (£p + {^y —}. 023) 

and this becomes a minimum when 



452 THERMODYNAMICS OF THE STEAM-ENGINE. 

becomes a minimum. Differentiating with regard to/ 7 , and 
equating the first differential coefficient to zero, gives 



/ = V> t A (324) 

Since the air is supplied to each cylinder at the tempera- 
ture t l9 their volumes should be inversely as the absolute 
pressures p 1 and/'. This method also leads to an equal dis- 
tribution of work between the two cylinders, for if the value 
of p' from equation (324) is introduced into equations (322) 
and (323) we shall obtain 

^,= ^=A^{g;') V -i}; • • (325) 

and the total work of compression is 



n — 1 

2« 



Three-stage Compressors. — When very high pressures 
are required, as where air is used for storing energy, it is cus- 
tomary to use a compressor with a series of three cylinders, 
through which the air is passed in succession, and to cool the 
air on the way from one cylinder to the next. If the initial 
and final pressures are /, and / a , and if p' and p" are the 
pressures in the intermediate receivers in which the air is 
cooled, the conditions for most economical compression may 
be deduced in the following way : 

The work of compressing one pound of air in the several 
cylinders will be 

^av^I^F-!}; • • • (327) 

n ( 1 p 



^.=/'«"^n 7>) " -'! (329) 






COMPRESSED AIR. 453 

But since the air is cooled to the initial temperature on 
its way from one cylinder to the other so that 

A^=/V=/V / ; (330) 

consequently the total work of compressing one pound of air 
will be 

W = W X +IV % + W 3 
This expression will be a minimum when 

becomes a minimum; that is, when 



n — 1 



dW\ n—\p' n n— 1 / 



A P 



and 



W')*>~ n >L-L n zn^j- ' ' V333; 

p> » p " » 

Equations (328) and (329) lead to 

/"=A/". (334) 

P"* =P'Pi'> (335) 

from which by elimination we have 

/ = Vp?p„ 00000. (336) 



454 THERMODYNAMICS OF THE STEAM-ENGINE. 

and 



P" = Vp,p; (337) 

Since the temperature is the same at the admission to each 
of the three cylinders, the volumes of the cylinders should 
be inversely proportional to the absolute pressures /„ p' , and 
p" . As with the compound compressors, this method of 
arranging a three-stage compressor leads to an equal distribu- 
tion of work between the cylinders. For, if the values of p' 
and p" from equations (336) and (337) are introduced into 
equations (327) to (329), taking account also of the equation 
(330) we shall have 

w x =w t = w, =Av>-^n { (j) ~^ - 1 } ; • (338) 

and consequently the total work of compression is 

W =Wtt\$T-*] (339) 

Friction and Imperfections. — The discussion has thus 
far taken no account of friction of the compressor nor of 
imperfections due to delay in the action of the valves and to 
heating the air as it enters the cylinder of the compressor. 

From comparisons of indicator-diagrams taken from the 
steam- and the air-cylinders of certain combined steam-engines 
and air-compressors at Paris, Professor Kennedy found a 
mechanical efficiency of 0.845. Professor Gutermuth found 
an efficiency of 0.87 for a new Riedler compressor. It will 
be fair to assume an efficiency of 0.85 for compressors which 
are driven by steam-engines; compressors driven by turbines 
will probably be affected to a like extent by friction. 

The following table given by Professor Unwin * shows the 
effect of imperfect valve-action and of heating the entering 

* Development and Transmission of Power, p. 182. 



COMPRESSED AIR, 



455 



air as deduced from tests on a Dubois-Francois compressor 
which had a diameter of 18 inches and a stroke of 48 inches: 

RATIO OF ACTUAL AND APPARENT CAPACITIES OF AN 

AIR-COMPRESSOR. 







Ratio of air 






delivered at 


Piston speed, 
feet per 


Revolutions 


a.mospheric 
pressure and 


per minute. 


temperature to 






volume dis- 






placed by 






piston. 


80 


IO 


O.94 


J 60 


20 


O.92 


200 


25 


O.90 


240 


30 


O.86 


280 


35 


O.78 



This table does not take account of the effect of clearance, 
nor is the clearance for the compressor stated. It is probable 
that five or ten per cent will be enough to allow for imperfect 
valve-action after the effect of clearance is properly calculated. 
The effect of clearance is to require a larger volume of cylinder 
than would be needed without clearance. The effect of 
imperfect valve-action and of heating of the entering air is to 
require an additional increase in the size of the cylinder of the 
air-compressor and also to increase the work of compression. 

Efficiency of Compression. — If air could 
be so cooled during compression that the 
temperature should not rise the compres- 
sion line cdj Fig. 89, would be an isothermal 
line and the work of compressing one pound Fig. 89. 

of air would be 

W = Pf>* + A*\ lo & ^ — Pf>x ; 




but p x v x = p^v 2 for an isothermal change, and consequently 



W=f> 1 v 1 log e ^-. 



(34o) 



456 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Some investigators have taken the work of isothermal 
compression, represented by equation (340), as a basis of 
comparison for compressors and have considered its ratio to 
the actual work of compression as the efficiency of compres- 
sion. This throws together into one factor the effect of 
heating during compression and the effect of imperfect valve- 
action. 

Professor Riedler* obtained indicator-diagrams from the 
cylinders of a number of air-compressors and drew upon them 
the diagrams which would represent the work of isothermal 
compression, without clearance or valve losses. A compari- 
son of the areas of the isothermal and the actual diagrams 
gave the arbitrary efficiency of compression just described. 
The following table gives his results: 



ARBITRARY EFFICIENCY OF COMPRESSION. 



Type of compressor. 



Colladon, St. Gothard 
do. 

Sturgeon 

Colladon 

Slide-valve 

Paxman 

Cockerill 

Riedler two-stage 



Pressures in 

main, 
atmospheres. 



Lost work in 

per cent of 

useful work. 



IO5.O 
92.O 
94-3 

33.15 

49-3 
42.7 
40.2 
12.07 



Arbitrary 
efficiency. 



O.488 
O.521 

0.5I5 
O.772 
O.670 
O.7OI . 

0-7I3 
O.892 



A similar comparison for a fluid piston-compressor showed 
an efficiency of 0.84. 

There are three notable conclusions that may be drawn 
from this table: (1) there is much difference between com- 
pressors working at the same pressures, (2) a simple com- 
pressor loses efficiency rapidly as the pressure rises, and (3) 
the compound or two-stage compressor shows a great advan- 
tage over a simple compressor. 



* Development and Distribution of Power, Unwin. 



COMPRESSED AIR. 457 

Test of a Blowing-engine. — Pernolet* gives the follow- 
ing test of a blowing-engine used to produce the blast for 
Bessemer converters at Creusot. The engine was a two- 
cylinder horizontal engine, with the cranks at right angles. 
The piston-rod for each cylinder extended through the 
cylinder-head and actuated a double-acting compressor. 
The dimensions were: 

Diameter, steam-pistons 1.2 metres. 

" air-pistons 1.5 " 

Stroke. 1.8 

Diameter of fly-wheel 8.0 " 

At 28 revolutions per minute the following results were 
obtained: 

Indicated horse-power of steam-cylinders. ... 1082 
" " " air-cylinders 999 

Efficiency 0.92 

Temperature of air admitted io° C. 

" " delivered 6o° C. 

Pressure of air delivered, metres of mercury 

above the atmosphere 1.2 1 

Pressure of air in supply-pipe, metres of mer- 
cury below the atmosphere 0.023 

At 25 revolutions there was no sensible depression of 
pressure in the supply-pipe. 

The air from such a blowing-engine probably suffers little 
loss of temperature after compression. 

Air-pumps. — The feed-water supplied to a steam-boiler 
usually contains air in solution, which passes from the boiler 
with the steam to the engine and thence to the condenser. 
In like manner the injection-water supplied to a jet-condenser 
brings in air in solution. Also there is more or less leakage 
of air into the cylinder communicating with the condenser 

* V Air Comprimd, 1876. 



458 THERMODYNAMICS OF THE STEAM-ENGINE. 

and into the exhaust-pipe or the condenser itself. An air- 
pump must therefore be provided to remove this air and to 
maintain the vacuum. The air-pump also removes the con- 
densed steam from a surface-condenser, and the mingled 
condensed steam and injection-water from a jet-condenser. 
If no air were brought into the condenser the vacuum would 
be maintained by the condensation of the steam by the 
injection, or the cooling water, and it would be sufficient to 
remove the water by a common pump, which, with a surface- 
condenser, might be the feed-pump. 

The weight of injection-water per pound of steam, calcu- 
lated by the method on page 250, will usually be less than 20 
pounds, but it is customary to provide 30 pounds of injection- 
water per pound of steam, with some method of regulating 
the quantity delivered. 

It may be assumed that the injection-water will bring in 
with it one-twentieth of its volume of air at atmospheric 
pressure, and that this air will expand in the condenser to a 
volume inversely proportional to the absolute pressure in the 
condenser. The capacity of the air-pump must be sufficient 
•to remove this air and the condensed steam and injection- 
water. 

An air-pump for use with a surface-condenser may be 
smaller than one used with a jet-condenser. In marine work 
it is common to provide a method of changing a surface- into 
a jet-condenser, and to make the air-pump large enough to 
give a fair vacuum in case such a change should become 
advisable in an emergency. 

Seaton * states that the efficiency of a vertical single- 
acting air-pump varies from 0.4 to 0.6, and that of a double- 
acting horizontal air-pump from 0.3 to 0.5, depending on the 
design and condition; that is, the volume of air and water 
actually discharged will bear such ratios to the displacement 
of the pump. 

* Manual of Marine Engineering. 



COMPRESSED AIR. 



459 



He also gives the following table of ratios of capacity of 
air-pump cylinders to the volume of the engine cylinder or 
cylinders discharging steam into the condenser: 

RATIO OF ENGINE AND AIR-PUMP CYLINDERS. 



Description of pump. 


Description of engine. 


Ratio. 


Single-acting vertical 

<< « < 

<< (t 

it << 

(i << 

Double-acting horizontal. . . 
< < i < 

<< << 

«< < « 

<< << 


Jet-condensing, expansion i£ to 2 
Surface- " " i|to2 
Jet- " " 3 to 5 
Surface- " " 3 to 5 

Jet-condensing, expansion i£ to 2 
Surface- " " i| to 2 
Jet- " " 3 to 5 
Surface- " " 3 to 5 
compound. .= 


6 to 8 

8 to 10 

10 to 12 

12 to 15 

15 to 18 
10 to 13 

13 to 16 

16 to 19 
19 to 24 
24 to 28 



Calculation for an Air-compressor. — Let it be required 
to find the dimensions of an air-compressor to deliver 300 
cubic feet of air per minute at 100 pounds per square inch by 
the gauge, and also the horse-power required to drive it. 

If it is assumed that the air is forced into the delivery- 
pipe at the temperature of the atmosphere, and, further, that 
there is no loss of pressure between the compressor and the 
delivery-pipe, equation (318) for finding the volume drawn 
into the compressor will be reduced to 

V, = V~ = 300 X L =2341 cubic feet. 

'A 14.7 

If now we allow five per cent for imperfect valve-action 
and for heating the air as it is drawn into the compressor the 
apparent capacity of the compressor will be 



2341 -r- 0.95 = 2464 cubic feet ; 

this is the volume on which the power for the compressor 
must be calculated. 

If the clearance of the compressor is 0.02 of the piston 



460 THERMODYNAMICS OF THE STEAM-ENGINE. 

displacement, then the factor for allowing for clearance will 
be 

m\pj ' m 100 \ 147 / 100 * JJ 

if the exponent of the equation representing the expansion of 
the air in the clearance is 1.4. Consequently the volume 
on which the dimensions of the compressor must be based is 

2464 — 0.9332 = 2640 cubic feet. 

At 80 revolutions per minute the mean piston displacement 
will be 

2640 -r(2X 80) =16.5 cubic feet. 

Assuming a stroke of 3 feet, the mean area of the piston 
must be 

(144 X 16.5) -~ 3 = 792 square inches. 

Allowing 16 square inches for a piston-rod 4J- inches in 
diameter gives a mean area of 800 square inches for the 
piston, which corresponds very nearly to 32 inches for the 
diameter of the piston. 

The power expended in the compressor-cylinder may be 
calculated by equation (315), using for V l the apparent 
capacity of the compresior, giving 



1.4 - 1 



= 144 X 14.7 X 2464 XM I W— ) 

33000 X (1.4- 1) (\M.7/ ) 

If the friction of the combined steam-engine and com- 
pressor is assumed to be 15 per cent the horse-power of the 
steam-cylinder must be 

442 ^-0.85 = 520. 

If the temperature of the atmosphere drawn into the 



COMPRESSED AIR. 461 

compressor is 70 F., then by an equation like (80), page 67, 
the delivery temperature will be 

T.= T l $f 1 = (46o. 7 + 7 o)(^p = 954 »4 

absolute, or about 494 ° F. 

The calculation has been carried on for a simple com- 
pressor, but there will be a decided advantage in using a 
compound compressor for such work. Such a compressor 
should have for the pressure in the intermediate reservoir 



p' — Vp^pi = V 1 14.7 X 14.7 = 41.06 pounds. 

The factor for allowing for clearance of the low-pressure 
cylinder will now be 

i -- ^r+- = 1 - — f- — V + — =0.9784. 

m\pj ' m 100 \ 14.7 ] 100 *' 

The loss from imperfect action of the valves and for heat- 
ing of the air as it enters the compressor will be less for a 
compound than for a simple compressor, but we will here 
retain the value 2464 cubic feet, previously found for the 
apparent capacity of the compressor. The volume from which 
the dimensions of the condenser will be found will now be 

2464 -J- 0.9784 = 2518 cubic feet, 

which with 80 revolutions per minute will give 15.74 cubic 
feet for the piston displacement, and 755.5 square inches for 
the effective piston area, if the stroke is made 3 feet, as 
before. Adding 16 inches for the piston-rod, which will be 
assumed to pass entirely through the cylinder, will give for 
the diameter of the low-pressure cylinder 3 if inches. 

Since the pressure p' is a mean proportional between /, 
and / 2 , the clearance factor for the high-pressure cylinder will 
be the same as that for the low-pressure cylinder, and, as the 



462 THERMO D YNAMICS OF THE STEAM-ENGINE, 

volumes are inversely proportional to the pressures/, and p' i 
the high-pressure piston displacement will be 

(15.74 X 14.7) -v- 41.06 = 5.64 cubic feet. 

If we allow 8 inches for a rod 4J inches in diameter at one 
side of the piston, then the mean area of the piston will be 
278.7 square inches, which corresponds to a diameter of 18-J 
inches for the high-pressure cylinder. In reality the piston- 
rod for the compound compressor may have a less diameter 
than the rod for a simple compressor, because the maximum 
pressure on both pistons will be less than that for the piston 
of the simple compressor. Again, the rod which extends 
from the large to the small piston may be reduced in size. 
But details like these which depend on the calculation of 
strength cannot properly receive much attention at this place. 
The power required to drive the compressor may be 
derived from equation (236), replacing ^,, the specific volume, 
by V v the apparent capacity of the low-pressure cylinder. 
Using the apparent capacity already obtained, 2464 cubic 
feet, we have for the power expended in the air-cylinders 



2 X 144 X 14-7 X 2464 X 1.4 ( /4i-o6 



1.4 



H - p -- 33000 x (1.4 - 1) i 1 14.7 r' 4 r f- 377; 

and, as before, allowing 15 per cent for friction of the engine 
and compressoi, we have for the indicated horse-power of the 
steam-engine 

377-- 0.85 =444- 

The temperature at the delivery from the low-pressure 
cylinder will be for 70 F. atmospheric temperature 

(460.7 + 70) (^r) "* =711° -7 

absolute, or 2 5 1 ° F. Since /' is a mean proportional between 
p x and p al this will also be the temperature of the air delivered 
by the high-pressure cylinder. 



COMPRESSED AIR. 4 6 3 

Friction of Air in Pipes. — The resistance to the flow of a 
liquid through a pipe is represented in works on hydraulics 
by an expression having the form 

u" I 
Z 2gm> (340 

in which C is an experimental coefficient, u is the velocity in 
feet per second, g is the acceleration due to gravity, / is the 
length of the pipe in feet, and m is the hydraulic mean depth, 
which last term is obtained by dividing the area of the pipe 
by its perimeter. For a cylindrical pipe we have consequently 



Di 



1 



7i d* -T- 7td =s \d. (34 2 ) 



The expression represents the head of liquid required to 
overcome the resistance of friction in the pipe when the 
velocity of flow is u feet per second. Such an expression 
cannot properly be applied to flow of air through a pipe when 
there is an appreciable loss of pressure, for the accompanying 
increase in volume necessitates an increase of velocity, whereas 
the expression treats the velocity as a constant. If, however, 
we consider the flow through an infinitesimal length of pipe, 
for which the velocity may be treated as constant, we may 
write for the loss of head due to friction 

u 2 dl 

2g m \o°tb) 

This loss of head is the vertical distance through which the 
air must fall to produce the work expended in overcoming 
friction, and the total work thus expended may be found by 
multiplying the loss of head by the weight of air flowing 
through the pipe. It is convenient to deal with one pound 
of air, so that the expression for the loss of head also repre- 
sents the work expended. 

The air flowing through a long pipe soon attains the tem- 
perature of the pipe and thereafter remains at a constant 
temperature, so that our discussion for the resistance of fric- 



464 THERMODYNAMICS OF THE STEAM-ENGINE. 

tion may be made under the assumption of constant tempera- 
ture, which much simplifies our work, because the intrinsic 
energy of the air remains constant. Again, the work done by 
the air on entering a given length dl will be equal to the 
work done by the air when it leaves that section, because the 
product of the pressure by the volume is constant. 

Since there is a continual increase of volume corresponding 
to the loss of pressure to overcome friction, and consequently 
a continual increase of velocity from the entrance to the exit 
end of the pipe, there is also a continual gain of kinetic 
energy. But the velocity of air in long pipes is small and the 
changes of kinetic energy can be neglected. 

The air expands by the amount dv as it passes through 
the length dl of pipe, and each pound does the work pdv. 
This work must be supplied by the loss of head, and, since 
there is no other expenditure of energy, the work expended 
in the loss of head is equal to the work done by expansion; 
consequently 

u* dl . 

pdv = C (343) 

r 2gm KJ ^ J/ 

But from the characteristic equation 

pv = RT (344) 



we have 



dv = — -^~d/>, 
P 



which substituted in equation (343) gives 

y dl RT j , . 

^m = --T dp (345) 

If the area of the pipe is A square feet, and if W pounds 
of air flow through it per second, then 

Wv WRT 
U = -A = -Ap-> ..... (346) 



COMPRESSED AIR. 465; 

in which v is the specific volume, for which a value may be 
derived from equation (344). Replacing u in equation (345) 
by the value just derived, we have 

W'T'R'dl _ RT 
C 2gAym •" J~ dp '' 

r W*dl p 7 
■■• ^l^n=-RT ap (347) 

Integrating between the limits L and o, and p^ and/,, we 
have 

w*l a'-?; , „, 

Yrf 5 ^ - RT (348> 

But from equation (346) the velocity at the entrance to the 
pipe where the pressure \s p x will be 

WRT A W A P^ 

U > = -ApT a " d W= -RT' 
so that equation (348) may be reduced to 

^gA'mR'T' RT ' 

. u t 'L A' -A' ,, v 

• • C ^7^ = ~JT~ (349) 

Equation (349 may be solved as follows: 

(gRTm A'-A')*. #,«\ 

*' = 1 "CZ a^~ J ! " • - - (35 } 

A'AJi-^ejs} ; (35«) 

c-^¥ <*> 

The first two forms allow us to calculate either the velocity 
or the loss of pressure; the last form may be used to calcu- 
late values of £ from experiments on the flow through pipes.. 



4^6 T HER MOD YNAMICS OF THE STEAM-ENGINE. 

From experiments made by Riedler and Gutermuth* 
Professor Unwin f deduces the following values for £: 

Diameter of pipe, feet. £ 



0.492 


0.00435 


0.656 


0.00393 


0.980 


0.00351 



For pipes over one foot in diameter he recommends for use 

C = 0.003. 

Replacing the hydraulic mean depth m by \d, its value 
for round pipes, and using R = 53.22 and g = 32. 16, we have 
in place of equation (351) 

*=M i_ JSrF-- •• • ■ • (353) 

All of the dimensions are given in feet, but from the form 
of the equation it is evident that the pressures may be in any 
convenient units, for example, in pounds per square inch 
absolute. 

For example, let us find the loss of pressure of 300 cubic 
feet per minute if delivered through a six-inch pipe a mile 
long, the initial pressure being 100 pounds by the gauge. 

The velocity of the air will be 

(300 -f- 60) - — = 5 -*- -^p- = 25.5 feet. 
The terminal pressure will consequently be 



c tju?L ) r 0.0044 X 25.5 X 5 2 8o ) * 

^ = ^r~«^f -ll4 ' 7 r — 430(460.7 + 70 )i 1 

= 106.8 pounds, 

* Neue Erfahrungen iiber die Kraftversorgung von Paris durch Druck- 
luft, 1891. 

f Development and Distribution of Power. 



COMPRESSED AIR. 



467 



with 70 F. for the temperature of the atmosphere and with 
Q = 0.0044. Consequently the loss of pressure is about eight 
pounds. 

Compressed-air Engines. — Engines for using com- 
pressed air differ from steam-engines only in details that 
depend on the nature of the working fluid. In some instances 
compressed air has been used in steam-engines without any 
change; for example, in Fig. 90 the dotted diagram was 




Fig. 90. 

taken from the cylinder of an engine using compressed air, 
and the dot-and-dash diagram was taken from the same end 
of the cylinder when steam was used in it. The full line ab 
is a hyperbola and the line ac is the adiabatic line for a gas; 
both lines are drawn through the intersection of the expansion 
lines of the two diagrams. 

Power of Compressed-air Engines. — The probable 
mean effective pressure attained in the cylinder of a com- 
pressed-air engine, or to be expected in a 
projected engine, may be found in the same 
manner as is used in designing a steam- 
engine. In Fig. 91 the expansion curve 
1 2 and the compression curve 3 o may be 
assumed to be adiabatic lines for a gas 
represented by the equation 

. K K 

pv =p x v x , 

and the area of the diagram may be found in the usual way, 
and therefrom the mean effective pressure can be determined. 




Fig. 91.; 



468 THERMODYNAMICS OF THE STEAM-ENGINE. 

Having the mean effective pressure, the power of a given 
engine or the size required for a given power may be deter- 
mined directly. The method will be illustrated later by an 
example. 

Air-consumption. — The air consumed by a given com- 
pressed-air engine may be calculated from the volume, 
pressure, and temperature at cut-off or release, and the 
volume, temperature, and pressure at compression, in the 
same way that the indicated consumption of a steam-engine 
is calculated ; but in this case the indicated and actual con- 
sumption should be the same, since there is no change of 
state of the working fluid. Since the intrinsic energy of a 
gas is a function of the temperature only, the temperature 
will not be changed by loss of pressure in the valves and 
passages, and the air at cut-off will be cooler than in the 
supply-pipe, only on account of the chilling action of the walls 
of the cylinder during admission, which action cannot be 
energetic when the air is dry, and probably is not very im- 
portant when the air is saturated. 

Final Temperature. — If the expansion in a compressed- 
air engine is complete, i.e., if it is carried down to the pres- 
sure in the exhaust-pipe, then, assuming that there are no 
losses of pressure in valves and passages, the final temperature 
may be found by the equation 



If the expansion is not complete, then the temperature at 
the end of expansion may be found by the equation 

T r = T,(j$"\ (355) 

in which V c is the volume in the cylinder at cut-off and V r at 
release, T r is the absolute temperature at the end of expan- 
sion, and 7" s is the temperature at cut-off, assumed to be the 



COMPRESSED AIR. 4 6 9 

same as in the supply-pipe. T r is not the temperature during 
back-pressure nor in the exhaust-pipe. When the exhaust- 
valve is opened at release the air will expand suddenly, and 
part of the air will be expelled at the expense of the energy 
in the air remaining — much as though that air expanded 
behind a piston, and the temperature in the cylinder during 
exhaust and at the beginning of compression may be calcu- 
lated by equation (354). The temperature in the exhaust- 
pipe will not be so low, for the temperature of the escaping 
air will vary during the expulsion produced by sudden expan- 
sion, and will only at the end of that operation have the 
temperature T if while the energy expended on that air to 
give it velocity will be restored when the velocity is reduced 
to that in the exhaust pipe. 

Volume of the Cylinder. — The determination of the 
volume of the cylinder of a compressed-air engine which uses 
a stated volume of air per minute is the converse of the 
determination of the air consumed by a given engine, and can 
be found by a similar process. We may calculate the volume 
of air, at the pressure in the supply-pipe, consumed per stroke 
by an engine having one unit of volume for its piston dis- 
placement, and therefrom find the number of units of volume 
of the piston displacement for the required engine. 

Interchange of Heat. — The interchanges of heat between 
the walls of the cylinder of a compressed-air engine and the 
air working therein are of the same sort as those taking place 
between the steam and the walls of the cylinder of a steam- 
engine; that is to say, the walls absorb heat during admission 
and compression, if the latter is carried to a considerable 
degree, and yield heat during expansion and exhaust. Since 
the walls of the cylinder are never so warm as the entering 
air nor so cold as the air exhausted, the walls may absorb 
heat during the beginning of expansion and yield heat during 
the beginning of compression. 

The amount of interchange of heat is much less in a com- 
pressed-air engine than in a steam-engine. With a moderate 



4?0 THERMODYNAMICS OF THE STEAM-ENGINE. 

expansion the interchanges of heat between dry air and the 
walls of the cylinder are insignificant. Moisture in the air 
increases the interchanges in a marked degree, but does not 
make them so large that they need be considered in ordinary 
calculations. 

Moisture in the Cylinder. — The chief disadvantage in the 
use of moist compressed air — and it is fair to assume that 
compressed air is nearly if not quite saturated when it comes 
to the engine — is that the low temperature experienced when 
the range of pressures is considerable causes the moisture to 
freeze in the cylinder and clog the exhaust-valves. The 
difficulty may be overcome in part by making the valves and 
passages of large size. Freezing of the moisture may be pre- 
vented by injecting steam or hot water into the supply-pipe 
or the cylinder, or the air may be heated by passing it through 
externally heated pipes or by some similar device. In the 
application of compressed air to driving street-cars the air 
from the reservoir has been passed through hot water, and 
thereby made to take up enough hot moisture to prevent 
freezing. The study of gas-engines suggests a method of 
heating compressed air which it is believed has never been 
tried. The air supplied to a compressed-air engine, or a part 
of the air, could be caused to pass through a lamp of proper 
construction to give complete combustion, and the products 
of combustion passed to the engine with the air. Should 
such a device be used it would be advisable that the tem- 
perature of the air should be raised only to a moderate degree 
to avoid destruction of the lubricants in the cylinder, and the 
combustion at all. hazards must be complete, or the cylinder 
would be fouled by unburned carbon. 

Compound Air-engines. — When air is expanded to a con- 
siderable degree in a compressed-air engine a gain may be 
realized by dividing the expansion into two or more stages in 
as many cylinders, provided that the air can be economically 
reheated between the cylinders. The heat of the atmosphere 
or of water at the same temperature may sometimes be used 



COMPRESSED A JR. 47 1 

for this purpose. It is not known that machines of this con- 
struction have been used. If they were to be constructed 
the practical advantages of equal distribution of work and 
pressure would probably control the ratio of the volumes of 
the cylinders. 

Calculation for a Compressed-air Engine. — Let it be 
required to find the dimensions for a compressed-air engine to 
develop ioo indicated horse-power at the pressure of 92 
pounds by the gauge and at 70 F. Assume the clearance to 
be five per cent of the piston displacement, and assume the 
cut-off to be at half stroke, the release to be at the end of 
the stroke, and the compression at one-tenth of the stroke. 

If the piston displacement is represented by D, then the 
volume in the cylinder at cut-off will be 0.30Z}, that at 
release will be 1.05Z}, and that at compression will be o. 15Z?. 
The absolute pressures during supply and exhaust may be 
assumed to be 106.7 an< ^ J 4«7 pounds per square inch. The 
work for one stroke of the piston will be 

„ , 144 X 106.7 X0.30ZM /0.30V-4- 1 ) 
W= 144X106.7x0.25^ + -^ -J-—^— j i-[~) [ 

_ 144 X 14-7 X0.15ZK /0.05V- 4-1 ) 
- 144 X 14.7 X o. 9 D _ J , _J_|) [ 

= 144^(26.68 + 31.530 — 13-23 — i-9 6 ) = 144 X 43-02.£>. 

The corresponding mean effective pressure is 43.02 pounds 
per square inch. If the engine is furnished with large ports 
and automatic valve-gear the actual mean effective pressure 
ma)/ be 0.9 of that just calculated, or 38.7 pounds per square 
inch. 

For a piston displacement D the engine will develop at 
150 revolutions per minute 



144 x 38.7^ X 2 X 150, 

— — — ^— horse-power ; 

33000 



472 THERMODYNAMICS OF THE STEAM-ENGINE. 

and conversely to develop ioo horse-power the piston dis- 
placement must be 

ioo X 33000 , . , 

D = = 1.974 cubic feet, 

144 x 387 X 2 X 150 y/ ^ 

and with a stroke of 2 feet the effective area of the piston 
will be 

I.974 X 144 -H 2 = 142. 1 square inches. 

If the piston-rod is 2 inches in diameter it will have an area 
of 3.14 square inches, so that the mean area of the piston will 
be 143.7 square inches, corresponding to a diameter of 13^ 
inches. 

We find, consequently, that an engine developing 100 
horse-power under the given conditions will have a diameter 
of 13J inches and a stroke of 2 feet, provided that it runs at 
150 revolutions per minute. 

In order to determine the amount of air used by the 
engine we must consider that the air caught at compression 
is compressed to the full admission-pressure of 106.7 pounds 
absolute. Part of this compression is done by the piston and 
part by the entering air, but for our present purpose it is 
immaterial how it is done. The volume filled by air at 
atmospheric pressure when the exhaust-valve closes (including 
clearance) is 0.15 of the piston displacement. When the 
pressure is increased to 106.7 pounds the volume will be 
reduced to 

o.isf-^M 1 ' 4 = 0.017 
\106.7J 

of the piston displacement. The volume drawn in from the 
supply-pipe will consequently be 

p.25 -f- 0.05 — 0.017 = 0.283 

of the piston displacement. If the compression occurred 
sufficiently early to raise the pressure to that in the supply- 
pipe before the admission-valve opened, then only 0.25 of the 



COMPRESSED AIR. 473 

piston displacement would be used per stroke and a saving 
of about 13 per cent would be attained; in such case the 
mean effective pressure would be smaller and the size of the 
cylinder would be larger. 

The air-consumption for the engine appears to be 

2 X 1 50 X 0.283 X pist. displ. = 2 X 1 50 X 0.283 X 1 .974 = 167.6 

cubic feet per minute. The actual air-consumption will be 
somewhat less on account of loss of pressure in the valves and 
passages; it may be fair to assume 160 cubic feet per minute 
for the actual consumption. 

In order to make one complete calculation for the use of 
compressed air for transmitting power the data for the com- 
pressed-air engine have been made to correspond with the 
results of calculations for an air-compressor on page 459 and 
for the loss of pressure in a pipe on page 466. Since there 
is a loss of pressure in flowing through the pipe at constant 
temperature, there is a corresponding increase of volume, so 
that the pipe delivers 

300 X 1 14.7 -5- 106.7 = 322.6 

cubic feet per minute. Our calculation for the air-consump- 
tion of an engine to deliver 100 horse-power gives about 160 
cubic feet, from which it appears that the system of com- 
pressor, conducting-pipe, and compressed-air engine should 
deliver 

100 X 322.6 4- 160 = 200 -(- horse-power. 

If the friction of the compressed-air engine is assumed to 
be ten per cent the power delivered by it to the main shaft 
(or to the machine driven directly from it) will be 

200 X .9 = 180 horse-power. 

The steam-power required to drive a simple compressor 
was found to be 520 horse-power; it consequently appears 
that 

180 -~ 520 = 0.34 



474 THERMODYNAMICS OF THE STEAM-ENGINE. 

of the indicated steam-power is actually obtained for doing 
work from the entire system of transmitting power. If, 
however, a compound compressor is used, then the indicated 
steam-power is 444, and of this 

1 80 H- 444 = 0.40 

will be obtained for doing work. 

If, however, we consider that the power would in any case 
be developed in a steam-engine, and that the transmission 
system should properly include only the compressor-cylinder, 
the pipe, and the compressed-air engine, then our basis of 
comparison will be the indicated power of the compressor- 
cylinder. For the simple compressor we found the horse- 
power to be 442, which gives for the efficiency of transmission 

180 -r- 442 = 0.41, 

while the compound compressor demanded only 377 horse- 
power, giving an efficiency of 

180 -r- 2,77 = 0.48. 

It appeared that the failure to obtain complete compression 
involved a loss of about 13 per cent in the air-consumption. 
It may then be assumed that with complete compression our 
engine could deliver 200 horse-power to the main shaft In 
that case the efficiency of transmission when a compound 
compressor is used may be 0.53. 

Efficiency of Compressed-air Transmission. — The pre- 
ceding calculation exhibits the defect of compressed air as a 
means of transmitting power. It is possible that somewhat 
better results may be obtained by a better choice of pressures 
or proportions; Professor Unwin estimates that when used 
on a large scale from 0.44 to 0.51 of the indicated steam- 
power may be realized on the main shaft of the compressed- 
air engine. On the other hand, when compressed air is used 
in small motors, and especially in rock-drills and other mining- 
machinery, much less efficiency may be expected. 



COMPRESSED AIR. 475 

Experiments made by M. Graillot * of the Blanzy mines 
showed an efficiency of from 22 to 32 per cent. Experiments 
made by Mr. Daniel at Leeds gave an efficiency varying from 
0.255 to 0.455, with pressures varying from 2.75 atmos- 
pheres to 1.33 atmospheres. An experiment made by Mr. 
Kraft f gave an efficiency of 0.137 for a small machine, using 
air at a pressure of five atmospheres without expansion. 

Compressed air has been used for transmitting power either 
where power for compression is cheap and abundant, or where 
there are reasons why it is specially desirable, as in mining and 
tunnelling. It is now used to a considerable extent for driving 
hand-tools, such as drills, chipping-chisels, and calking-tools, 
in machine- and boiler-shops, and in shipyards. It is also used 
for operating cranes and other machines where power is used 
only at intervals, so that the condensation of steam (when 
used directly) is excessive, and where hydraulic power is liable 
to give trouble from freezing. 

Compressed air has been used to a very considerable extent 
for transmitting power in Paris. The system appears to be 
expensive and to be used mainly on account of its convenience 
for delivering small powers or in places where the cold exhaust 
can be used for refrigeration. The trouble from freezing of 
moisture in the cylinder has been avoided by allowing the 
air to flow through a coil of pipe which is heated externally by 
a charcoal fire. Professor Unwin estimates that an efficiency of 
transmission of 0.75 may be attained under favorable condi- 
tions when the air is heated near the compressor, but he does 
not include the cost of fuel for reheating in this estimate. 

Storage of Power by Compressed Air. — Reservoirs or 
cylinders charged with compressed air have been used to store 
power for driving street-cars. A system developed by Mekar- 
ski uses air at 350 to 450 pounds per square inch in reservoirs 
having a capacity of 75 cubic feet. The car also carries a tank 



* Pernolet, V Air Comprime, pp. 549, 550. 

f Revue universelle des Mines, 2 serie, tome vi. 



47^ THERMODYNAMICS OF THE STEAM-ENGINE. 

of hot water at a temperature of about 350 F., through which 
the air passes on the way to the motor and by which it is 
heated and charged with steam. This use of hot water gives 
a secondary method of storing power and also avoids trouble 
from freezing in the motor-cylinders, Air at much higher 
pressures has been used for driving street-cars in New York 
City, but the particulars have not been given to the public. 

The calculation for storage of power may be made in much 
the same way as that for the transmission of power; the chief 
difference is due to the fact that the air is reduced in pressure 
by passing it through a reducing-valve on the way from the 
reservoir to the motor. By the theory of perfect gases such 
a reduction of pressure should not cause any change of tem- 
perature, but the experiments of Joule and Thomson (page 72) 
show that there will be an appreciable, though not an impor- 
tant, loss of temperature when there is a large reduction of 
pressure. Thus at 70 F. or 2i°.i C. the loss of temperature 
for each 100 inches of mercury will be 

°°'9 2 X (Hi)' = °°-79 C = if F. 

Now 100 inches of mercury are equivalent to about 49 
pounds to the square inch, so that 100 pounds difference of 
pressure will give about 3J- F. reduction of temperature and 
1000 pounds difference of pressure will give about 35 F. re- 
duction of temperature. The last figures are far beyond the 
limits of the experiments and the results are therefore crude. 
Again, the air in passing through the reducing-valve and the 
piping beyond will gain heat and consequently show a smaller 
reduction of temperature. The whole subject of loss of 
temperature due to throttling is more curious than useful and 
need not be considered in practical calculations. 

For an example of the calculation for storage of power let 
us find the work required to store air at 450 pounds per square 
inch in a reservoir containing 75 cubic feet. Replacing the 
specific volume v i in equation (339) by the actual volume, we 



COMPRESSED AIR. 477 

have for the work of compression (not allowing for losses and 
imperfections) 

W= 3 X 14-7 X 144 X 75 -^-{f 4 -^) 3 ^- 1} 

1.4— 1 l\ 14.7/ J 

= 649300 foot-pounds. 

If the pressure is reduced to 50 pounds by the gauge before 
it is used the volume of air will be 

75 X 464.; -*- 64.7 = 539 cubic feet. 

The work for complete expansion of one pound to the pressure 
of the atmosphere will be 

K — I I V 3 / 

and the work for 539 cubic feet will be 

1.4- \ (\A.7\ i-4 / 

144 X 64. 7 X 5 39^^ J 1 - (g^J [ - 487300 

foot-pounds, without allowing for losses or imperfections. 
The maximum efficiency of storing and restoring energy by 
the use of compressed air in this case is therefore 

487300 ^ 649300 = 0.75. 

In practice the efficiency cannot be more than 0.50, if 
indeed it is so high. 

It may not be out of place to call attention to a danger 
that may arise if air at high pressure is suddenly let into a 



1.4- 1 



478 THERMODYNAMICS OF THE STEAM-ENGINE. 

pipe which has oil mingled with the air in it or even adher- 
ing to the side of the pipe. The air in the pipe will be com- 
pressed and its temperature may become high enough to 
ignite the oil and cause an explosion. That this danger is 
not imaginary is shown by an explosion which occurred under 
such conditions in a pipe which was strong enough to with- 
stand the air-pressure. 



CHAPTER XVIII. 
REFRIGERATING-MACHINES. 

A REFRIGERATING-machine is a device for producing low- 
temperatures or for cooling some substance or space. It may 
be used for making ice or for maintaining a low temperature 
in a cellar or storehouse. 

Refrigeration on a small scale may be obtained by the 
solution of certain salts ; a familiar illustration is the solution 
of common salt with ice, another is the solution of sal am- 
moniac in water. Certain refrigerating-machines depend on 
the rapid absorption of some volatile liquid, for example, of 
ammonia by water; if the machine is to work continuously 
there must be some arrangement for redistilling the liquid 
from the absorbent. The most recent and powerful refriger- 
ating-machines are reversed heat-engines. They withdraw 
the working substance (air or ammonia) from the cold-room 
or cooling-coil, compress it, and deliver it to a cooler or con- 
denser. Thus they take heat from a cold substance, do work 
and add heat, and finally reject the sum of the heat drawn in 
and the heat equivalent of the work done. These reversed 
heat-engines, however, are very far from being reversible 
engines, not only on account of imperfect valve action and 
losses of pressure, but because the walls of the compressor- 
cylinder have an appreciable effect on the working substance. 

Two forms of refrigerating-machines are in common use, 
air refrigerating-machines and ammonia refrigerating-machines. 
Sometimes sulphur dioxide or some other volatile fluid is used 
instead of ammonia. 

479 



480 



THERMODYNAMICS OF THE STEAM-ENGINE. 



Air Refrigerating-machine. — The general arrangement 
of an air refrigerating-machine is shown by Fig. 92. It con- 
sists of a compression-cylinder A, an expansion-cylinder B of 
smaller size, and a cooler C. It is commonly used to keep 
the atmosphere in a cold-storage room at a low temperature, 
and has certain advantages for this purpose, especially on ship- 
board. The air from the storage-room comes to the com- 
pressor at or about freezing-point, is compressed to two or 
three atmospheres and delivered to the cooler, which has the 
same form as a surface-condenser, with cooling water entering 
at e and leaving at f. The diaphragm mn is intended to im- 
prove the circulation of the cooling water. From the cooler 
the air, usually somewhat warmer than the atmosphere, goes 
to the expansion-cylinder i?, in which it is expanded nearly 




Wi > 



Fig. 92. 



to the pressure of the air and cooled to a low temperature, 
and then delivered to the storage-room. The inlet-valves 
a, a and the delivery-valves b> b of the compressor are moved 
by the air itself; the admission-valves c, c and the exhaust- 
valves d, d of the expansion-cylinder are like those of a steam- 
engine and must be moved by the machine. The difference 
between the work done on the air in the compressor and that 
done by the air in the expansion-cylinder, together with the 
friction work of the whole machine, must be supplied by a 
steam-engine or other motor. 



REFRIGERA TING- MA CHINES. 48 1 

It is customary to provide the compression-cylinder with 
a water-jacket to prevent overheating, and frequently a spray 
of water is thrown into the cylinder to reduce the heating and 
the work of compression. Sometimes the cooler C, Fig. 92, 
is replaced by an apparatus resembling a steam-engine jet-con- 
denser, in which the air is cooled by a spray of water. In 
any case it is essential that the moisture in the air, as well as 
the water injected, should be efficiently removed before the 
air is delivered to the expansion-cylinder; otherwise snow will 
form in that cylinder and interfere with the action of the 
machine. Various mechanical devices have been used to col- 
lect and remove water from the air, but air may be saturated 
with moisture after it has passed such a device. The Bell- 
Coleman Company use a jet-cooler with provision for collect- 
ing and withdrawing water, and then pass the air through 
pipes in the cold-room on the way to the expansion-cylinder. 
The cold-room is maintained at a temperature a little above 
freezing-point, so that the moisture in the air is condensed 
upon the sides of the pipes and drains back into the cooler. 
The same machine as made by Menck and Hambrock is pro- 
vided with a device for removing moisture from the air that 
is shown by Fig. 93. Air from the cooler comes in by the 
pipe a, is distributed by the annular perforated pipe b , and 
passes out to the expansion-cylinder by the pipe c. The 
chamber E is surrounded by a jacket through which passes 
the cold air on the way from the expansion-cylinder to the 
cold-room. Since the air in the jacket is many degrees below 
freezing-point, the walls of the chamber E are quickly covered 
with frost, which accumulates till a considerabfe thickness is 
attained ; afterwards the moisture condenses and runs down 
to the bottom of the chamber, from whence it is withdrawn. 
A coil of steam-pipe dd is provided for thawing ice and snow 
that may accumulate at the bottom of the chamber. Since 
the same air is used continuously, being taken from the cold- 
room, chilled, and returned, the effect of these devices is to 
remove the moisture from the air in the cold-room and to 



482 



THERMODYNAMICS OF THE STEAM-ENGINE. 



maintain a cold, dry atmosphere in it, which is well adapted 
to preserving all kinds of perishable provisions. 

When an air refrigerating-machine is used as described the 
pressure in the cold-room is necessarily that of the atmos- 
phere, and the size of the machine is large as compared with 
its performance. The performance may be increased by run- 
ning the machine on a closed cycle with higher pressures ; for 
example, the cold air may be delivered to a coil of pipe in a 




Fig. 93. 



non-freezing salt solution, from which the air abstracts heat 
through the walls of the pipe and then passes to the com- 
pressor to be used over again. The machine may then be used 
to produce ice, or the brine may be used for cooling spaces or 
liquids. A machine has been used for producing ice on a 
small scale, without cooling water, on the reverse of this prin- 
ciple : that is, atmospheric air is first expanded and chilled 
and delivered to a coil of pipe in a salt solution, then the air 
is drawn from this coil, after absorbing heat from the brine, 
compressed to atmospheric pressure, and expelled. 



REFRIGERA TING- MA CHINES. 48 3 

Proportion of Air of Refrigerating-machines. — The per- 
formance of a refrigerating-machine may be stated in terms 
of the number of thermal units withdrawn in a unit of time, 
or in terms of the weight of ice produced. The latent heat 
of fusion of ice may be taken to be 80 calories or 144 B. T. u. 

Let the pressure at which the air enters the compression- 
cylinder be p x , that at which it leaves be / 3 ; let the pressure 
at cut-off in the expanding-cylinder be p % and that of the back- 
pressure in the same be/ 4 ; let the temperatures correspond- 
ing to these pressures be /. , £,, t s , and / 4 , or, reckoned from 
the absolute zero, 7",, 7!,, T 2 , and T t . With proper valve- 
gear and large, short pipes communicating with the cold- 
chamber^ may be assumed to be equal to p^ and equal to 
the pressure in that chamber. Also t x may be assumed to be 
the temperature maintained in the cold-chamber, and / 3 may 
be taken to be the temperature of the air leaving the cooler. 
With a good cut-off mechanism and large passages p % may be 
assumed to be nearly the same as that of the air supplied to 
the expanding-cylinder. Owing to the resistance to the pas- 
sage of the air through the cooler and the connecting pipes 
and passages, p 3 is considerably less than/,. 

It is essential for best action of the machine that the ex- 
pansion and compression of the expanding-cylinder shall be 
complete. The compression may be made complete by set- 
ting the exhaust-valve so that the compression shall raise 
the pressure in the clearance-space to the admission-pres- 
sure p s at the instant when the admission-valve opens. The 
expansion can be made complete only by giving correct 
proportions to the expanding- and compression-cylinders. 

The expansion in the expanding-cylinder may be assumed 
to be adiabatic, so that 



r< - fM (356) 



T. W 



484 



THERMODYNAMICS OF THE STEAM-ENGINE. 




Were the compression also adiabatic the temperature / s 

b could be determined in a similar manner; 

but the air is usually cooled during com- 

_c pression, and contains more or less vapor, so 

that the temperature at the end of com- 

Fig. 94. pression cannot be determined from the 

pressure alone, even though the equation of the expansion 

curve be known. 

Let the air passing through the refrigerating-machine per 
minute be M; then the heat withdrawn from the cold-room is 

Q x = Mc p {t x - t A ) (357) 

The work of compressing M pounds of air from the pres- 
sure^ to the pressure/, in a compressor without clearance is 

(Fig- 94) 

W c = M\p^ + I 'pdv-p^A; 



W c = M\p 9 v t + 



P& 

n — 1 






1 - ij-A^iL 



W c = M\ p x v, 



£1 
A 



+ 



n 



P 



iUV\ 



— 1 



- A^ [ , 



w <=^>Mw " - 



(358) 



provided that the compression curve can be represented by an 
exponential equation. If the compression can be assumed to 
be adiabatic 



W £ = Mp 1 v i 



K 



P, 



K- I ( V>, 

for in such case we have the equations 



- 1 1 = ¥h[f % - O; (359) 






AR = c — c„ — c. 



K — I 



K 



REFRIGERA TING- MA CHINES. 48 5 

If the expansion is complete in the expanding-cylinder, as 
should always be the case, then the equation for the work 
done by the air will have the same form as equation (358) or 
{359), replacing t x and/ by / 4 and/ 4 , and t % and/ a by t 3 and 
j> 3 ; so that 



*r=Jfr A _2L.{(*)" -1} (360) 

and for adiabatic expansion 

W e = ^(t,-t.) (361) 

The difference between the works of compression and ex- 
pansion is the net work required for producing refrigeration ; 
consequently 

W=W C -W.= *£\t, -/,-/, + /.}; . (362) 
■or, replacing M by its value from equation (357), 

w =z t ' +t t \Z t ;~ t ' -- (3 6 3) 

The net horse-power required to abstract Q x thermal units 
per minute is consequently 

Pn = 7j*Q L t. + t<-t x -t 

33000 t t - t K u ^ 

where t x is the temperature of the air drawn into the com- 
pressor and £, is the temperature of the air forced by the com- 
pressor into the cooler, and t % is the temperature of the air 
supplied to the expanding-cylinder and t A is the temperature 
of the cold air leaving the expanding-cylinder. The gross 
horse-power developed in the steam-engine which drives the 
refrigerating-machine is likely to behalf again as much as the 
net horse-power or even larger. The relation of the gross 



486 THERMODYNAMICS OF THE STEAM-ENGINE. 

and the net horse-powers for any air refrigerating-machine 
may readily be obtained by indicating the steam- and air- 
cylinders, and may serve as a basis for calculating other 
machines. 

The heat carried away by the cooling water is 

Q,= Q, + AW. (365) 

If compression and expansion are adiabatic, then 

<2, = Mc p {f x - / 4 + f, + / 4 - ^ - t s ) = Mc p (t, - t 3 ) ; (366) 

or, replacing M by its value from equation (357), 

&= Q'f^r- •.••'•• (367) 

If the initial and final temperatures of the cooling water are 
t 4 and t ky and if % snd Q k are the corresponding heats of the 
liquid, then the weight of cooling water per minute is 

G — ^ Q t 2~lJl . . (368) 

The compressor-cylinder must draw in M pounds of air per 
minute at the pressure p % and the temperature t lf that is, with 
the specific volume v x ; consequently its apparent piston dis- 
placement without clearance will be, at N revolutions per 
minute, 

M, MfiT,, ...... (369> 

2N 2Np x 

for the characteristic equation gives 

Replacing M by its value from equation (357), we have 

D = & RT * (370) 

^ 2Nc p pit x - Q U/ ' 



REFRIGERA TING-MA CHINES. 4%7 

Since all the air delivered by the compressor must pass 
through the expanding-cylinder, its apparent piston displace- 
ment will be 

D e =D c P ^ (37i) 

If , the pressure of the air entering the compression-cylin- 
der, is equal to / 4 , that of the air leaving the expanding-cylin- 
der (as may be nearly true with large and direct pipes for car- 
rying the air to and from the cold-room) equation (371) will 
reduce to 

D e = D c ^ (372) 

Both the compressor- and the expanding-cylinder will have 
a clearance, that of the expanding-cylinder being the larger. 
As is shown on page 447, the piston displacement for an air- 
compressor with a clearance may be obtained by dividing the 
apparent piston displacement by the factor 

If the expansion and compression of the expanding-cylin- 
der are complete the same factor may be applied to it. For 
a refrigerating-machine n may be replaced by k for both cyl- 
inders. To allow for losses of pressure and for imperfect 
valve action the piston displacements for both compressor- 
and expanding-cylinders must be increased by an amount 
which must be determined by practice ; five or ten per cent 
increase in volume will probably suffice. In practice the ex- 
pansion in the expanding-cylinder is seldom complete. A 
little deficiency at this part of the diagram will not have a 
large effect on the capacity of the machine, and will prevent 
the formation of a loop in the indicator-diagram ; but a large 
drop at the release of the expanding-cylinder will diminish 
both the capacity and the efficiency of the machine. 



488 THERMODYNAMICS OF THE STEAM-ENGINE. 

The temperature t K and the capacity of the machine may 
be controlled by varying the cut-off of the expanding-cylin- 
der. If the cut-off is shortened the pressure / 2 will be in- 
creased, and consequently T 4 will be diminished. This will 
make D e , the piston displacement of the expanding-cylinder, 
smaller. A machine should be designed with the proper pro- 
portions for its full capacity, and then, when running at re- 
duced capacity, the expansion in the expanding-cylinder will 
not be quite complete. 

Calculation for an Air-compression Machine. — Required 
the dimensions and power for an air refrigerating-machine to 
produce an effect equal to the melting of 200 pounds of ice 
per hour. Let the pressure in the cold-chamber be 14.7 
pounds per square inch and the temperature 32 ° F. Let the 
pressure of the air delivered by the compressor-cylinder be 
39.4 pounds by the gauge or 55.1 pounds absolute, and let 
there be ten pounds loss of pressure due to the resistance of the 
cooler and pipes and passages between the compressor- and 
the expanding-cylinder. Let the initial and final temperatures 
of the cooling water be 6o° F. and 8o° F., and let the 
temperature of the air coming from the cooler be 90 F. Let 
the machine make 60 revolutions per minute. 

With adiabatic expansion and compression the tempera- 
tures of the air coming from the compressor- and discharged 
from the expanding-cylinder will be 

T % = 492.7 (~) M =7H-9 i ••; / 3 =254°.2F. 

r 4 = 460.7 + 90 (I±7) M = 402.3 ; .-. t< = - 5 8°. 4 F. 

\44- 1/ 

The melting of 200 pounds of ice is equivalent to 
200 X 144 -7- 60 = 480 B. T. U. 



REFRIGERA TING-MA CHINES. 489 

per minute; consequently the net horse-power of the machine 
is by equation (364) 

P - 7780, ti + t<-t x -t t 
n ' 33000 t x - t K 

_ 778 X 480 254.2 — 58.4— 32 — 90 
33000 32 + 58.4 

778 X 480 X 73-8 



33000 X 9°-4 



= 9 .2 4 H. P., 



and the indicated power of the steam-engine may be assumed 
to be 14 horse-power. 

By equation (370) the apparent piston displacement of the 
compressor without clearance will be 



a- &RT > 



2Nc p pit, - / 4 ) 
480 x 53-22 x 492.7 



2 x 60 x 0.2375 x H4 x 14.7 (32 + 58.4) 



= 2.33 cu. ft. 



By equation (372) the apparent piston displacement of the 
expanding-cylinder without clearance will be 

T t 402 . 3 

D. — D c ^ = 2.33 X = i.QO cubic feet. 

c T x JJ 492.7 * 

If the clearance of the compressor-cylinder is 0.02 of its 
piston displacement, then the factor for clearance by equation 
(316) is 

1 • 1 

1 /AV* l 2 /54. i\"m , 2 

1 - - - ) + - = 1 (^— ) H = 0.979, 

mxpj ' m 100 \ 14. 7/ 100 ^'^ 

so that the piston displacement becomes 

2-33 -J- 0.979 = 2.38 cubic feeto 



490 



THERMODYNAMICS OF THE STEAM-ENGINE. 



If, further, the clearance of the expander-cylinder is 0.0$ 
of its piston displacement the factor for clearance becomes 



5 /44-iV 4 ,5 „ ~ 
- 1 +— ,= °-9 6 3; 



f— ) 



100V14.7 



100 



which makes the piston displacement 

1.90 -f- 0.963 = 1.97 cubic feet. 

If now we allow ten per cent for imperfections we will 
get for the dimensions : stroke 2 feet, diameter of the com- 
pressor-cylinder 15^ inches, and diameter of the expanding- 
cylinder 14 inches. 

Compression Refrigerating-machine. — The arrangement 
of a refrigerating-machine using a volatile liquid and its vapor 
is shown by Fig. 95. The essential parts are the compressor 
A, the condenser £, the valve D, and the vaporizer C. The 
compressor draws in vapor at a low pressure and temperature, 
compresses it, and delivers it to the condenser, which consists 
of coils of pipe surrounded by cooling water that enters at e 




Fig. 95. 

and leaves at f. The vapor is condensed, and the resulting 
liquid gathers in a reservoir in the bottom, from whence it is 



REFRIGERA TING-MA CHINES. 49 r 

led by a small pipe having a regulating-valve D to the vapor- 
izer or refrigerator. The refrigerator is also made up of coils 
of pipe, in which the volatile liquid vaporizes. The coils may 
be used directly for cooling spaces, or they may be immersed 
in a tank of brine, which may be used for cooling spaces or for 
making ice. Fig. 95 shows the compressor with one single- 
acting vertical cylinder which has head-valves, foot-valves, 
and valves in the piston. Single-acting compressors com- 
monly have two cylinders ; horizontal compressors usually 
have one double-acting cylinder. Some vertical compressors 
are double-acting. 

The cycle which has been stated for the compression 
refrigerating-machine is incomplete, because the working fluid 
is allowed to flow through the expansion-cock into the expand- 
ing-coils without doing work. To make the cycle complete 
there should be a small expanding-cylinder in which the liquid 
could do work on the way from the condenser to the vaporiz- 
ing-coils; but the work gained in such a cylinder would be 
insignificant, and it would lead to complications and diffi- 
culties. 

Proportions of Compression Refrigerating-machines. — 
The liquid condensed in the coils of the condenser flows to the 
expansion-cock with the temperature t x and has in it the heat 
q^ In passing through the expansion-cock there is a partial 
vaporization, but no heat is gained or lost. The vapor flow- 
ing from the expansion-coils at the temperature t^ and the 
pressure p^ is usually dry and saturated, or perhaps slightly 
superheated, as it approaches the compressor. Each pound 
consequently carries from the expanding-coils the total heat A 2 . 
Consequently the heat withdrawn from the expanding-coil 
by a machine using M pounds of fluid per minute is 

Q> = M(\ % -q x ) (374) 

The compressor-cylinder is always cooled by a water- 
jacket, but it is not probable that such a jacket has much 



49 2 THERMODYNAMICS OF THE STEAM-ENGINE. 

effect on the working substance, which enters the cylinder dry 
and is superheated by compression. We may consequently 
calculate the temperature of the vapor delivered by the com- 
pressor by aid of equation (1 8 1), page 135, giving 

T,= TjM h (374) 

As has already been pointed out, the vapor approaching 
the compressor may be treated as though it were dry and 
saturated, each pound having the total heat A 2 . The vapor 
discharged by the compressor at the temperature t s and the 
pressure p x will have the heat 

W.-^ + K 

The heat added to each pound of fluid by the compressor is 
consequently 

c P (t s - O + A, - Jl 9 , 

and an approximate calculation of the horse-power of the 
compressor may be made by the equation 

' 33000 ' ' ' '■J'-'-' 

•or, substituting for M from equation (374), 

P <~ 33000(A - ?,) ' • • {i7) 

The power thus calculated must be multiplied by a factor 
to be found by experiment in order to find the actual power 
of the compressor. Allowance must be made for friction to 
find the indicated power of the steam-engine which drives the 
motor; for this purpose it will be sufficient to add ten or fif- 
teen per cent of the power of the compressor. 

The heat in the fluid discharged by compressor is equal to 
the sum of the heat brought from the vaporizing-coils and the 



REFRIGERA TING-MA CHINES. 493 

heat equivalent of the work of the compressor. The heat 
that must be carried away by the cooling water per minute is 
consequently 

Q, = M{\ -q x )+M\ cjt, - t x ) + A, _ X, ; 
.-. Q, = Jf\c#.~ Q + r,\ (376) 

where r x is the heat of vaporization at the pressure p x . 

If the cooling water has the initial temperature t d and the 
final temperature t k , and if Q t and Q k are the corresponding 
heats of the liquid for water, then the weight of cooling 
water used per minute will be 

G = Mc ,(t-y + rt (377) 

If the vapor at the beginning of compression can be as- 
sumed to be dry and saturated, then the volume of the piston 
displacement of a compressor without clearance, and making 
N strokes per minute, is 

*-# 0* 

To allow for clearance, the volume thus found may be 
divided by the factor 

, _ utx + L 

m \pj m 

as is explained on page 447. The volume thus found is further 
to be multiplied by a factor to allow for inaccuracies and 
imperfections. 

The vapors used in compression-machines are liable to be 
mingled with air or moisture, and in such case the performance 
of the machine is impaired. To allow for such action the size 
and power of the machine must be increased in practice above 



494 THERMOD YNAM1CZ OF THE STEAM-ENGINE. 

those given by calculation. Proper precautions ought to be 
taken to prevent such action from becoming of importance. 

Calculation for a Compression Refrigerating-machine. — 
Let it be required to find the dimensions and power for an 
ammonia refrigerating-machine to produce 2000 pounds of ice 
per hour from water at 8o° F. Let the temperature of the 
brine in the freezing-tank be 15 F., and the temperature in 
the condenser be 85 F. Assume that the machine will have 
one double-acting compressor, and that it will make 80 revolu- 
tions per minute. 

The heat of the liquid at 8o° F. is 48.09 B. T. U., and the 
heat of liquefaction of ice is 144, so that the heat which must 
be withdrawn to cool and freeze one pound of water will be 

48.09 -|- 144 = 192.09 B. T. u. 

If we allow 50 per cent loss for radiation, conduction, and 
melting the ice from the freezing-cans, the heat which the 
machine must withdraw for each pound of ice will be about 
300 B. T. U. ; consequently the capacity of the machine will 
be 

Q x = 2000 x 300 -7- 60 = 10000 B. T. u. per minute. 

The pressures corresponding to 15 and 85 ° F. are 42.43 
and 165.47 pounds absolute per square inch, so that by equa- 
tion (374) 

T,= T^y=^ + <6o. 7) (^) : 668. 5 . 

.-. t s = 668.5 —460.7 = 207°.8 F. 
The-horse-power of the compressor is 

33000(A 2 -^) 
778 x 10000)0.50836(207.8 — 85)4-556- 535} 

= 7 £7 =S 41.2. 

33000(535 - 58) 



REFRIGERA TING-MA CHINES. 



495 



If we allow 10 per cent for imperfections the compressor 
will require 45 horse-power. If further 15 per cent is allowed 
for friction the steam-engine must develop 53 horse-power. 

From equation (374) the weight of ammonia used per 
minute is 

M= 1 -r-(A. a — 00 = 10000-^(535 - 58) = 21 pounds; 

and by equation (378) the piston displacement for the com- 
pressor will be 



D = 



N 



21 X 6.Q3 ,-. r 

— — = o.qi cubic feet. 

2 X 80 y 



If ten per cent is allowed for clearance and imperfect valve 
action the piston displacement will be one cubic foot, and the 
diameter may be made 10J inches and the stroke 20 inches. 

Fluids Available. — The fluids that have been used in 
compression refrigerating-machines are ether, sulphurous acid, 
ammonia, and a mixture of sulphurous acid and carbonic acid, 
known as Pictet's fluid. The pressures of the vapors of these 
fluids at several temperatures, and also the pressure of the 
vapors of methylic ether and carbonic acid, are given in the 
following table : 

PRESSURES OF VAPORS, MM. OF MERCURY. 



Temperatures, 

degrees 

Centigrade. 


Ether. 


Sulphur 
Dioxide. 


Methyl- 
ether. 


Ammonia. 


Carbon 
Dioxide. 


Pictet's 
Fluid. 


- 30 




287.5 


576.5 


866.1 




585 


— 20 


68.9 


479-5 


882.O 


1392. 1 


15142 


745 


— IO 


114 7 


762.5 


1306.6 


2144.6 


20340 


1018 


O 


184.4 


1165.1 


1879.O 


3183.3 


26907 


1391 


IO 


286.8 


1719.6 


2629.O 


4574-0 


34999 


1938 


20 


432.8 


2462 . 1 


3586.0 


6387.8 


44717 


2584 


30 


634-8 


3431-8 


4778.O 


8701.0 


56119 


3382 


40 


907.0 


4670.2 




II595-3 


69184 


4347 



Ether was used in the early compression-machines, but at 
the temperatures maintained in the refrigerator the pressure is 
small and the specific volume large, so that the machines, like 
air refrigerating-machines, were either feeble or bulky. More- 



49^ THERMODYNAMICS OF THE STEAM-ENGINE. 

over, air was liable to leak into the machine and unduly heat the 
compressor cylinder. Sulphur dioxide has been used success- 
fully, but it has the disadvantage that sulphuric acid may be 
formed by the leakage of moisture into the machine, in which 
case rapid corrosion occurs. Ammonia has been extensively 
used in the more recent machines with good results. When 
distilled from an aqueous solution it is liable to contain con- 
siderable moisture. As is shown by the table, Pictet's fluid 
has a pressure at low temperature intermediate between the 
pressures of sulphur dioxide and ammonia, and the pressure 
increases slowly with the temperature. It has been used by 
the inventor only, and does not appear in practice to have any 
advantage over ammonia. 

Absorption Refrigerating Apparatus. — Fig. 96 gives an 
ideal diagram of a continuous absorption refrigerating appara- 
tus. It consists of the following essential parts: (1) the 
generator B, containing a concentrated solution of ammonia 
in water, from which the ammonia is driven by heat ; (2) the 
condenser C, consisting of a coil of pipe in a tank, through 
which cold water is circulated ; (3) the valve V, for regulating 
the pressures in C and in /; (4) the refrigerator /, consisting 
of a coil of pipe in a tank containing a non-freezing salt solu- 
tion ; (5) the absorber A, containing a dilute solution of 
ammonia, in which the vapor of ammonia is absorbed ; and 
(6) the pump P for transferring the solution from the bottom 
of A to the top of B; there is also a pipe connecting the bot- 
tom of B with the top of A. It is apparent that the condenser 
and refrigerator or vaporizer correspond to the parts B and C 
of Fig. 95, and that the absorber and generator take the place 
of the compressor. The pipes connecting A and B are 
arranged to take the most concentrated solution from A to 
By and to return the solution from which the ammonia has 
been driven, from B to A. In practice the generator B is 
placed over a furnace, or is heated by a coil of steam-pipe, to 
drive off the ammonia. Also, arrangements are made for 
transferring heat from the hot liquid flowing from B to A to 



REFRIGERA TING-MA CHINES. 



497 



the cold liquid flowing from A to B. As the ammonia is dis- 
tilled from water in B the vapor driven off contains some 
moisture, which causes an unavoidable loss of efficiency. 

The earliest absorption apparatus, made by Carre, con- 
sisted of a cylindrical receptacle containing a solution of 
ammonia, and acting alternately as generator and absorber, in 
open communication through a pipe with a vessel of double 




Fig. q6. 

conical form, acting alternately as condenser and refrigerator. 
In use, the generator was placed on a furnace and the con- 
denser in a tank of cold water, and the ammonia driven off 
from the solution condensed between the inner and outer 
conical surfaces of the condenser. When a sufficient amount 
of liquid ammonia had collected, the vessel containing the 
solution was transferred from the furnace to the cold-water 
tank, and became thereby changed into the absorber. The 
condenser at the same time became the vaporizer or refriger- 
ator, and after receiving a mould containing water to be frozen, 
was securely wrapped with non-conducting material. Appa- 
ratus of this kind is only fitted for work on a small scale, and 
is inefficient. 

An adaptation of Carry's apparatus has been used in re- 
frigerator-cars for carrying perishable freight. In the car are 
placed two receptacles — one containing liquid ammonia, which 
maintains a low temperature by vaporization ; and the other 



49 8 THERMODYNAMICS OF THE STEAM-ENGINE. 

containing water, to absorb the ammonia as it formed. At 
the end of the route, or when necessary, the receptacles are re- 
charged — one with liquid ammonia and the other with fresh 
water. The ammonia in the rejected solution is regained by 
distillation. 

Vacuum Refrigerating Apparatus. — A form of absorption 
apparatus uses water for the volatile liquid and concentrated 
sulphuric acid for the absorbent. From the fact that vapor of 
water at freezing-point has a very low tension such apparatus 
are called vacuum apparatus. 

The first apparatus of this kind was designed for freezing 
water in carafes, and consisted of a good air-pump and a re- 
ceptacle containing oil of vitriol. The carafe, well wrapped in 
non-conductor, was attached to a pipe leading to the sulphuric- 
acid receptacle, the pump was worked till a good vacuum was 
produced, and the acid was stirred to present fresh acid to the 
vapor which rapidly streamed from the water at the low pres- 
sure produced. The vaporization of about one-sixth of the 
weight of the water was' found to be sufficient to freeze the 
remainder. 

An ideal sketch of a continuous vacuum apparatus is shown 
by Fig. 97. At B is an air-pump capable of producing a 
vacuum of one or two mm. of mercury in the chamber AC. 
At //"there is a tank of concentrated sulphuric acid, from which 
a spray is delivered at J. The acid absorbs the vapor found 
in the chamber at the low pressure existing there, gathers in 
the tank J, and flows out through the pipe K, which is of suf- 
ficient length to deliver the acid against atmospheric pressure 
in the tank L. The dilute acid is reconcentrated and returned 
to the tank H. At G is a pipe supplying fresh water which 
passes through the water-injector s and throws a jet of salt 
solution into the chamber at A. The finely divided jet loses 
fresh water by vaporization, is chilled, and gathers in the bot- 
tom of the chamber. The salt solution flows through the pipe 
F in the cold-chamber EE, taking up heat on the way, and is 
again thrown into the chamber with a fresh supply of water 



REFRIGERA TING-MA CHINES. 



499 



from the pipe G. At iVand iV are screens to prevent splash- 
ing of water into the upper part of the chamber. 




a--^^mm^. 






'///)//////, /WW/// 



^ 



'////,//////////////////////////„ '<&, 



5l \ 



■ 



# 



^ 



v^/ 



i 
i 



Fig. 97. 



Tests of an Air Refrigerating-machine. — An air refriger- 
ating-machine, constructed under the Bell-Coleman patent, 
was tested by Professor Schroter* at an abattoir in Hamburg, 
where it was used to maintain a low temperature in a storage- 
room. The machine is horizontal, and has the pistons for the 
expansion- and compression-cylinders on one piston-rod, the 
expansion-cylinder being nearer the crank. Power is furnished 
by a steam-engine acting on a crank at the other end of the 
main shaft and at right angles to the crank driving the air- 
pistons. Both the steam-cylinder and the expansion-cylinder 
have distribution slide-valves, with independent cut-off valves. 
The main dimensions are given in the following table : 



* Untersuchungen an Kaltcmachinen, 1887. 



5oo 



THERMODYNAMICS OF THE STEAM-ENGINE. 



DIMENSIONS BELL-COLEMAN MACHINE. 



Diameter of piston, cm 

" " piston-rod, cm 

Stroke, m 

Clearance, per cent of piston displacement 



Steam 
Cylinder. 



Head 
end. 



53 
8.1 

0.605 
5-9 



Crank 
end. 



S3 
6.9 
0.605 
5-8 



Compression 
Cylinder. 



Head 
end. 



71 
j.o 
5.605 
c.4 



Crank 
end. 



71 
6.8 
0.605 
1.4 



Expansion 
Cylinder. 



Head 
end. 



53 
9.0 
0.605 
8.9 



Crank 
end. 



53 
9.0 
0.605 
8.9 



Water is sprayed into the compression-cylinder, and the 
air is further cooled by passing through an apparatus resem- 
bling a steam-engine jet-condenser, after which it is dried by 
passing it through a system of pipes in the cold-room before 
it passes to the expansion-cylinder. 

In the tests indicators were attached to each end of the 
several cylinders, and the temperature of the air was taken at 
entrance to and exit from each of the air-cylinders. Speci- 
mens of the indicator-diagrams from the air-cylinders show 
for the compressor a slight reduction of pressure during ad- 
mission and some irregularity during expulsion, and for the 
expansion-cylinder a little wire-drawing at cut-off, and a good 
expansion and compression, though neither is complete. No 
attempt was made to measure the amount and temperatures 
of the cooling water. 

The data and results of the tests and the calculations are 
given in Table LI. 

Tests of Compression-machines. — In Table LII are given 
the data and results of tests on three refrigerating-machines 
on the Linde system using ammonia, and of a machine on 
Pictet's system using Pictet's fluid, all by Professor Schroter. 
The tests on machines used for making ice were necessarily of 
considerable length, but the tests on machines used for cool- 
ing liquids were of shorter duration. 

The cooling water when measured was gauged on a weir or 
through an orifice. In the tests 3 to 6 on a machine used for 
cooling fresh water the heat withdrawn was determined by 



RE FRIG ERA TING-MA CHINES. 



50I 



Table LI. 

TESTS ON BELL-COLEMAN MACHINE. 



Number of Test 

Duration in hours 

Revolutions per minute 

Temperatures of air, degrees Centigrade: 

At entrance to compression cylinder , 

At exit from " 

At entrance to expansion " 

At exit from " " 

Mean effective pressure, kgs. per sq. cm.: 

Steam-cylinder: head end 

crank end 

Compression cylinder: head end 

crank end 

Expansion cylinder: head end 

crank end 

Indicated horse-power: 

Steam-cylinder 

Compression cylinder 

Expansion cylinder 

Mean pressure during expulsion from compression cylinder,:kgs. 
Mean pressure during admission to expansion cylinder, kgs.. .. 

Difference 

Calculation from compression diagram: 

Absolute pressure at end of stroke, kgs 

Absolute pressure at opening of admission-valve, kg. : 

Headend 

Crank end 

Volume at admission, per cent of piston displacement: 

Headend 

Crank end 

Weight of air discharged per stroke, kg.: 

Headend 

Crank end 

Weight of air discharged per revolution, kg 

Calculation from expansion diagram: 
Absolute pressure at release, kgs.: 

Headend 

Crank end 

Absolute pressure at compression, kgs.: 

Headend 

Crank end , 

Volume at release, per cent of p. d.: 

Head end 

Crank end 

Volume at compression, per cent of p. d.: 

Headend 

Crank end 

Air used per stroke, kg.: 

Head end 

Crank end 

Air used per revolution 

Difference of weights, calculated by compression and expan- 
sion diagrams, kg 

In per cent of the former 

Mean weight of air per revolution, kg 

Elevation of temperature at constant pressure, degrees Centi- 
grade 

Heat withdrawn per H. P. per hour, calories 



I. 

6 
65.05 


II. 

1.63 
61.2 


III. 
2.93 
63-5 


!9-3 

27-3 

19.0 

-47.0 


17-5 

26.8 

16.6 

- 47.0 


19. 1 
27.2 
19. 1 

— 47 


2.263 
2.239 
1.900 
1.869 
1.592 
1. 615 


2-336 
2.294 
1. 861 
1.825 
1.589 
1-594 


2-343 
2.301 
1.870 
1.906 
1.626 
1.624 


85.12 

128.85 

60.10 

3-35 
2.82 

o-53 


82.35 

"8.55 

56.12 

3-25 
2.83 
0.42 


85-7i 
126.01 

59-46 
3-40 
2.84 
0.56 


1.04 


1.04 


1 .04 


0.783 
0.765 


0.788 
0.749 


0.764 
0.765 


6.15 
8.50 


5-95 
8.41 


6.03 
7.91 


0.2744 
0.2716 
0.546 


0.2764 
0.2742 
o-55i 


0.2750 
0.2730 

0.548 


1.32 
i-45 


i-3i 
1.44 


I.-33 

1 .46 


1. 14 
1.20 


1. 14 
1. 19 


1. 17 
1.22 


104.65 
106. 1 


104.7 
106.3 


104.8 
106.4 


16.5 
19.8 


16.0 
19.6 


16.6 
20.6 


0.234 
0254 
0.488 


0.233 
0.254 
0.487 


0.238 
0.255 
°-493 


0.058 
10.6 


0.064 
11. 6 


0.055 
10. 


o-5H 


0.519 


0.520 


66.3 
37i 


64- 5 
354 


66.1 
363 



taking the temperatures of the water cooled, and by gauging 
the flow through an orifice, for which the coefficient of flow 
was determined by direct experiment. The heat withdrawn in 
the tests 7 and 8 was estimated by comparison with the tests 
3 to 6. The net production of ice in the tests 1 and 2 was 



502 



THERM 0D YNAMICS OF THE STEAM-ENGINE. 



W 
►J 
« 

H 






X 

u 

< 

6 

H 

< 

2 -5 

3 2 



to 

H 

C/) 

W 



<» 



s 



5 



O - 
co w 



H 

:o 

w 

u 

CO 



CO 

W 
3 

en 

o 2 






Mean effective 
pressure com- 
pressor, kgs. per 
sq. cm. 


•pU3 31UBJ2) 


. ■*- to 
• 00 ■* 


VO N 

VO vo 


M tl OO Oi N 
IO M CM tT t~. 


00 H 


fT) »0 


IO »o 


io ro ■<*■ m m 


M N 


"PUS PB3H 


O CM 


CM «*- 

■vt- -VI- 


vo oo ■* ro io 
•* O o ro »o 


w 
t^ Ov 


■*- to 


VO to 


vo ro ■*- h i-i 


M W 


uossajd 
-ujod '3;nuim jad suoimioAa^i 


CO OO tv 


M H 


IO 

VO M 00 N t^ 


IO O 


-* o ■«*• 
vo to to 


vo Ov 
IO IO 


On vo io ■*- •* 
•«- vo vo vo vo 


vo vo 


•japUIjAo UIB31S 

jo i3A\od-9SJoq paTeaipuj 


VO M 




. H «0 N IO 


CM 


ro vo 
to vo 




• VO -4- M T»- 

(N ro o> Ov 


On • 
ON 


Mean effective 

pressure. Steam 

cylinder,kgs. per 

sq. cm. 


*PU3 31UBJ3 








ro 

VO 


M CM 




N Ct 


CM 


*PU3 PB3H 


M OS ■ 




M M 


■>*- 








CM 


•auiSua 
mB9is 'ainuim jad suopnioAa-jj 


O O 






IO 
vo vo 


•JS3J JO UOIlBJnQ 


C .. .. 

a 

VO * *> 

ro ro ° 


vo 
IO -*■ 


O 

jd oo 

IO IO 

ro 

ro ro oo W 

H 


ro O 
00 O 

ON -*" 


a 

o . 
o v- 

CD 

%% 

■g.g 

o.o 

'<" 52 

c "> 

Si " 

5 


•rata l 33ioj;s 




v : m 
io ■*• 


3 3 


2 S S 8 S 

ON 


2 2 


•mm 
'poj-uojsid jo aajaareiQ 


ro - to 

vo ■* to 


2 2 


.. ^ IO ^ 
- - - VO - 


2 2 


•ram 'aojsid jo jajaureiQ 


to o 
cm 2 to 
ro cm 


2 - 


o 

2 2 2 ro 2 


2 2 


a 

<u 
en 
I) 

-Sc 

u 

o c 

S ^ 
.2 ° 
'3 
c 
u 

a 
S 


•mm 'ajjojis 


CM 
00 " vS 


2 2 



2 -1-2 O 2 
r~ on 


2 2 


•rain 
'poj-uojsid jo jaianreiQ 


to 

to - 
>o 




io - vo - 


2 2 


•ram 'uojsid jo jaiaurEiQ 


IT) 

M . o 

_ 2 o 

ro 


2 : 


„ 
2 ro 2 vo 2 

ro -*■ 


2 2 


•uopBDiiddv 


Ice-making. 

j Cooling fresh ) 
( water. f 


- - 


C - he 

c - .s 

§ .s 

u 


2 2 


•auiqDBcu aqj jo ma^sAg 


u 

■a - - 

a - - 

3 


: 2 


- - (J " 


2 2 






•jaqain^ 


h eq M 


■* iO 


«B r- 0D CS O 

iH 





REFRIGERA TING -MA CHINES. 



503 



<3 



-1 

< 



s 






























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fi. 


unoq J3d j3Mod 




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N 


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rn 
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m 

on 


co 
m 


09 

in 


-ssjoq joss3jdtao3 js j 




m 


PI 


m 
m 


O 

rn 




VO 



m 


vo 


m 

(M 


VO 
<N 


OV 


2.2 






























H 


Ov 


Ov 


O 


r-^ 


On 





■* 


* 


0-1 


m 


Ov 


•jnoq 


rn 


O 


m 


N 


t-^ 


O 


co 


r-i 


r^ 


TT 


IN 


w 




VO 


M 


m 


VO 


t>» 


-Tn 


VO 


vo 







N 


Oi 


rt " 


jsd uMBjpqjiM ;b3^ 


(V) 


■+ 


m 
00 


Ov 


Ov 
00 





vO 


(J 

CO 


00 


•* 




VO 


E 
















































































O 4) 








m 


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-<h 




1^. 










m 


C 




<* 


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Ov 


rn 


(V) 


r^ 


O 




M 





r^ 





2ts 


•}JX3 5v 


■* 


m 


M 


N 


N 


•* 


Ov 


■*■ 


CO 


O 


Ov 


VO 


3J3-Q 




1 


| 










| 


| 


« 


<~> 


| 


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1 


1 


























































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n 










m 


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-+ 


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10 




c< 





r^ 





6 cd 


•sdubjius iy 


■<*• 


ifl 


M 


H 


M 


M 


Ov 


m 


on 





Ov 


VO 


4/ > 




1 


1 


H 


M 


w 


M 


1 


1 


T 


1 


1 


1 




.-SOIBl 




to 














M 


VO 


Ov 


m 




33u jnoq J3a usmoq 




H 






. 








m 


rs 


in 


00 


C 

3 
O 


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m 
















N 


CN 


" 


'SOfl3[ 




CO 














no 





(VI 


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•S 


'ssojS 'jnoq jsd 'jaMod 




■*■ 




. 










vo 


10 


(10 





a 
■a 

V 


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en 














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CN 


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cv 


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IT) 














Ov 
00 


VO 
(Tv 


rs 


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3 


'jnoq jsd jonpojd lsjij 


vO 

M 
















r^ 


cr, 


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m 


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a 


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O 


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t> 




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vO 


in 


u 


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00 


in 














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CO 


in 


•3 sssjSsp 'psqddns 





m 














m 


m 


m 


m 




J3113AV jo sjniEjsdrasj, 


o> 


00 


















M 


M 












Ov 




m 


O 


M 


in 


in 


in 


00 




•anoq 








H 




m 


m 


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^r 


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u-) 




00 




J3d ABMB P3UJB3 JB3H 








rn 




a- 



VO 



00 


CO 


O^ 


VO 


m 
t-» 












H 










M 


M 


[V| 


M 








VO 


00 


■*■ 





M 


LO 




in 







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& 

buo 








m 


m 







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c> 


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M 


m 


rn 


vn 


N 


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M 








« 


N 


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M 




m 




























c 








Ov 








hs 


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in 














n 


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m 








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w 




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t> 


rs 


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rn 












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VO 


00 


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m 



m 


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ts f 


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(VI 


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M 


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■*• •*• 


■* ■<«■ 


■* ■*■ 


•<*-■*« 


m 





M 


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p~ 

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in 

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a- 




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IN 


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c 










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H 


VO 


VO 


M 


CO 


r^ 


M 


N 


n 


N 


H 




o> 


m 


VO 


O 


H 


r^ 


aa 


■* 


W 


m 


VO 


M 


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crt 


■>*■ 


■* 


rn 


r^ 





m 


m 


m 


in 


D. u 




























i» Q. 




























3 ui 




























0° 






on 










rn 


rei 


t~» 




m 




•uoisjndxa 




m 










M 


vo 


r^ 


M 


N 


CO 


< 


Suunp josssjdraoo uj 




Ov 










CO 





m 


^j- 


■* 


in 








r-> 





m 


Ov 






M 





N 


(V! 


•JOSS3JO" 




O 


n 


m 


M 


"♦ 


M 


00 





ts 


* 





-aioD jo j3M.od-3s.ioq paiBoipuj 




in 


vo 

M 


to. 


Ov 


<* 

01 


CO 


M 


10 


VO 


VO 
VO 


in 








•aaquin^ 


F^ 


61 


CO 


«* 


>9 


CP 


r^» 


cc 


CS 


O 
1— 1 


H 


1— 



504 THERMODYNAMICS OF THE STEAM-ENGINE. 

determined directly ; and in the test 2 the loss from melting 
during the removal from the moulds was found by direct ex- 
periment to be 8.45 per cent. By comparison with this the 
loss by melting in the first test was estimated to be 7. J per 
cent. The gross production of ice in the refrigerator was cal- 
culated from the net production by aid of these figures. In the 
tests 9 to 12 on the Pictet machine the gross production was 
determined from the weight of water supplied, and the net 
production from the weight of ice withdrawn. 

A separate experiment on the machine used for cooling 
brine gave the following results for the distribution of power: 

Total horse-power. 57. 1 

Power expended on compressor . . . , 19.5 

" " " centrifugal pump 9.8 

" " " water-pump 3.6 

The centrifugal pump was used for circulating the brine 
through a system of pipes used for cooling a cellar of a brew- 
ery. The water-pump supplied cooling water to the con- 
denser and for other purposes. 

A similar test on the Pictet machine gave : 

Power of engine alone 7.9 H. P. 

" " " and intermediate gear. . .. 16.6 " 
" " " gear, and pump 20.0 " 

From the above data the following table was arranged for 
the several tests on this machine : 

INDICATED AND EFFECTIVE WORK. 



Number of Test. 



Indicated work without compressor 

** " engine alone 

Effective work of steam-engine 

Indicated work of steam-engine 

Mechanical efficiency of steam-engine 

Power absorbed by intermediate gearing 

Power absorbed by compressor 

Indicated power of compressor 

Mechanical efficiency of compressor 



9 


10 


11 


12 


19.9 


20.0 


20.0 


19.9 


7-9 


8.0 


7 


9 


7 


9 


77.1 


80.1 


84 


6 


94 


4 


91.2 


94-5 


99 


2 


109 


8 


0.84 


0.85 





«S 





8 


11. 2 


11. 2 


11 


2 


11 


2 


65. 9(?) 


68.9 


73 


4 


83 


2 


52.0 


61.7 


66 


4 


7S 





o. 7 9(?) 


0.89 





90 


o.c 


jo 



REFRIGERA TING- MA CHINES. 



505 



In 1888 comparative tests were made by Professor Schro- 
ter on a Linde and on a Pictet refrigerating-machine, in a 
special building provided by the Linde Company which had 
every convenience and facility for exact work. The following 
table gives the principal dimensions of the machines: 

PRINCIPAL DIMENSIONS OF LINDE AND PICTET 
REFRIGERATING-MACHINES. 



Diameter of steam-cylinder, cm 

do. compressor-cylinder, cm 

do. steam piston-rod, cm 

do. compressor-rod, cm 

Stroke of steam-piston, cm 

do. compressor, cm 

Diameter of pipe in vaporizers, external, mm. . 
do. internal, mm.. 

Length of pipe in first vaporizer, m 

do. second vaporizer, m 

Diameter of pipe in condenser, external, mm. . 

do internal, mm . . 

Length of pipe in condenser, m , 



Linde. 



Pictet. 



30.55 


31-63 


25.03 


28.6 


4-85 


5 


5-5 


5 


70 


62 


42 


62 


40- 5 


44 


32 


36 


556-5 


538.2 


558.5 


538.2 


38.5 


44 


30 


36 


556.2 


483.1 



The Linde machine used ammonia and was allowed to 
draw a mixture of liquid and vapor into the compressor, so 
that no water-jacket was required. The Pictet machine used 
Pictet's fluid and had the compressor cooled by a water-jacket. 

The data and results of the tests are given in Tables LIII 
and LIV. Five tests on each machine were made with only 
one of the two vaporizers in use, and three were made with 
both in use. The temperature of the salt solution or brine, 
from which heat was withdrawn by the vaporizers, was allowed 
to vary about three degrees centigrade from entrance to exit. 
The entrance temperatures were intended to be 6° C, —2° C, 
— io° C, and — 18 C. The cooling water was supplied to 
the condenser at about o,°.5 C. for all tests, and for all but 
one it left the condenser with a temperature of nearly 20 C. ; 
the fifth test on each machine was made with the exit tem- 
perature of the cooling water at about 35 C. 



506 



THERMO D YNAMICS OF THE STEAM-ENGINE. 







o 






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508 THERMODYNAMICS OF THE STEAM-ENGINE. 

The pressure in the compressor depended, of course, on 
the temperatures of the brine and the cooling-water. For all 
the tests except the fifth on each machine, the maximum 
pressure of the working substance was nearly constant : about 
9 kilograms per square centimetre for ammonia and about 4 
kilograms for Pectet's fluid. The fifth test had considerably- 
higher pressure, corresponding to the higher temperature in 
the condenser. The minimum pressure of the working sub- 
stance of course diminished as the brine temperature fell. 

The heat yielded per hour to the ammonia in the vapor- 
izer was calculated by multiplying together the amount of 
brine used in an hour, the specific heat of the brine, and its 
increase of temperature. But the initial and final tempera- 
tures were not quite constant, and so a correction was ap- 
plied as indicated in the tables. The heat abstracted from the 
ammonia in the condenser was calculated from the water used 
per hour and its increase of temperature. The calculation 
for Pectet's machine involves also the jacket-water and its in- 
crease of temperature. A correction is applied for the varia- 
tions of initial and final temperatures of the cooling-water. 
If the heat equivalent of the work of the compressor is added 
to the heat yielded by the vaporizer the sum should be equal 
to the heat abstracted by the cooling-water. The per cent 
of difference between these two calculations of the heat ab- 
stracted by the cooling-water is a measure of the accuracy of 
the tests. 

The refrigerative effect is obtained by dividing the heat 
yielded by the vaporizer by the horse-power of the steam-cyl- 
inder. The first four tests with constant temperature in the 
condenser show a regular decrease in the refrigerative effect 
for each machine as the temperature of the brine and the 
minimum pressure of the working substance is reduced. The 
three tests with the entire vaporizing surface in use show a 
like result. The fifth test, with a higher temperature in the 
condenser, shows a less refrigerative effect than the second test, 
which has nearly the same brine temperatures. These results 



REFRIGERA TING-MA CHINES. 



509 



are in concordance with the idea that a refrigerating-machine 
is a reversed heat-engine ; for a heat-engine will have a higher 
efficiency and will use less heat per horse-power when the 
range of temperatures is increased, and per contra, a refriger- 
ating-machine will be able to transfer less heat per horse- 
power as the range of temperatures is increased. 

Table LV gives the data and results of tests made by 
Professor Denton on an ammonia refrigerating-machine. The 
only items requiring explanation are the refrigerative effect 

Table LV. 

TESTS ON AMMONIA REFRIGERATING-MACHINE. 

By Professor J. E. Denton, Trans. Am. Soc. Meek. Eugr., vol. xii, p. 326. 



Pressure above atmosphere, pounds per square inch: 

Ammonia from compressor 

Ammonia back-pressure : 

Barometer, inches of mercury 

Temperature, degrees Fahrenheit: 

Brine, inlet . . 

do. outlet 

Condensing-water, inlet 

do. outlet 

Jacket-water, inlet 

Ammonia-vapor leaving brine-tank 

do. entering compressor 

do. leaving compressor 

do. do. calculated 

do. entering condenser 

Brine, pounds per minute 

Specific gravity 

Specific heat 

Ammonia, lbs. per min. by metre 

do. from compressor displacement. . 

Heat account, b. t. u. per minute: 

Given to ammonia by brine 

do. compressor 

do. atmosphere 

Total received by ammonia 

Taken from ammonia by condenser 

do. jackets 

do. atmosphere 

Total taken from ammonia 

Error, per cent 

Power, etc.: 

Revolutions per minute 

Horse-power steam-cylinder 

do. compressor 

Mechanical efficiency . . 

Refrigerative effect : 

Tons of ice (melted) in 24 hours.. 

B. T. U. abstracted from brine per horse-power, minutes 
Pounds of ice (melted) per pound of coal 



151 

28 

30.07 

36.76 
28.86 
44-65 
83.66 
44.65 
34-2 

39 

213 

229 

200 

2281 

1. 163 

0.82 



14776 

27860 

140 

17702 

17242 

608 

182 

18032 



58.09 
85.0 
65.7 
0.81 

74.8 

174 
24.1 



II 



152 
8.2 

29-59 

6.27 
2.03 

56.65 

85.4 

56.7 

14.7 

25 
263 

3°4 
218 

2173 
1. 174 
0.78 
14.68 



71876 

2320 

147 

9653 

9056 

712 

338 

10106 

5 

57-7 
71.7 

54-7 
0.83 

36.43 
197 
14. 1 



III 



147 

13 

29.87 

14.29 
2.29 
46.9 
85.46 
46.9 

3-° 
10.13 

239 

260 

209 

942.8 

1. 174 

0.78 

16.67 

22.10 

8824 

2518 

167 

1 1 409 

9910 

656 

250 

10816 

3-5 

57.88 
73-6 
59-4 
0.86 

44.64 

197 

17.27 



IV 



161 



45 
00 
86 

3 
2 

34 
221 

237 
168 

2374 
1. 174 
0.783 
28.32 
34 -5 1 

14647 

3020 

141 

17708 

17359 
406 
252 

18017 



58.89 
88.6 
71.2 
0.83 

74-56 
196 

23.37 



5IO THERMODYNAMICS OF THE STEAM-ENGINE. 

and the calculated temperature of the vapor leaving the con- 
denser ; the latter was calculated by the equation 

T T IPX'' 4 

and shows both the cooling effect of the jacket and the error 
in assuming an adiabatic compression. The refrigerative ef- 
fect was obtained by dividing the B.T.U. given to the am- 
monia in a minute by the horse-power of the steam-cylinder. 
The tons per horse-power in 24 hours was obtained by multi- 
plying the refrigerative effect in thermal units per minute by 
the number of minutes in a day and then dividing the product 
by 2000 (the pounds in a short ton) and by 144 (the heat of 
melting a pound of ice). The pounds of ice per pound of 
coal was based on an assumed consumption of three pounds 
of coal per horse-power per hour, and was calculated by mul- 
tiplying the B.T.U. per horse-power per minute by 60 and 
dividing by 3 X 144. 

The main dimensions of the machine were : 

Diameter of ammonia cylinder (single-acting) 12 inches. 

Stroke of ammonia cylinder (single-acting) 30 " 

Diameter of steam-cylinder 18 " 

Stroke of steam-cylinder 36 " 

Diameter of pipe for vaporizer and condenser 1 " 

Length of pipe in vaporizer 8000 feet. 

do. condenser 5000 " 

Test of an Absorption-machine. — The principal data 
and the results of a test made by Professor J. E. Denton* 
on an absorption ammonia refrigerating-machine are given in 
Table LVI. The machine is applied to chill a room of about 
400,000 cubic feet capacity at a pork-packing establishment 
at New Haven, Conn. In connection with this test the 
specific heat of the brine, which served as a carrier of heat 
from the cold room to the ammonia, was determined by direct 

* Trans. Am. Soc. Mech. Eng., vol. x, May, 1889. 



REFRIGERA TING-MA CHINES. 



5" 



Table LVI. 

TEST OF AN ABSORPTION-MACHINE. 
Seven Days' Continuous Test, Sept. 11-18, 18 



. f Generator. 

Average pressures Steam 

above atmosphere^ Cooler -;;; 

in lbs. per sq. in. [ Absorber . 



Average tempera- 
tures in Fahren- 
heit degrees. 



Brine 



Atmosphere in vicinity of machine. 
Generator 

j Inlet 

\ Outlet , 

^ j ( Inlet 

Condenser j Qmlet 

Absorber j^V;- 



i Upper outlet to generator 
Lower " " absorber 
Inlet from absorber 

Inlet from generator 

Water returned to main boilers from steam 
coil 



Average range ot 
temperatures 
Fahr. degrees. 



Condenser. 
Absorber. . 
Brine 



Brine circulated per j Cubic feet, 
hour. \ Pounds. . . 



Specific heat of brine 

Cooling capacity of machine in tons of ice per day of 24 hours. . 

Steam consumption per hour, to volatilize ammonia, and to 

operate ammonia pump lbs. 



British 
units. 



t h erm al 



• , j Per pound of brine. 



Eliminated ^ 

( lotal per hour 

Of refrigerating effect per pound of steam 

consumption 

D . , j At condenser, per hour 

Ke j ectea } At absorber « 

fOn entering genera- 
ls At* J tor COil 

Per pound of steam -< ~ , 

r On leaving genera- 

[_ tor coil 

Consumed by generator per lb. of steam 

condensed 



Condensing water per hour, in lbs 

Equivalent ice production per pound of coal, if one pound of 

coal evaporates ten pounds of steam at boiler 

Calories, refrigerating effect per kilogramme of steam consumed 

. ... . , ( Condensing coil 

Approximate c o 1 1 ) Absorber * ., 

surface in sq. ft. ] Steam (( 



150.77 
47.70 
23.69 
23-4 

80 
272 
21.205 
16.16 

54i 

80 

80 
in 
212 
178 
132 
272 

260 

25i 

5-13 

1,633.7 
119,260 

0.800 
40.67 

1,986 

4.104 

481,260 

243 

918,000 

1,116,000 

1,203 

271 

932 
36,000 

17. 1 

135 
870 
35o 
200 



512 THERMODYMAMICS OF THE STEAM-ENGINE. 

experiment. The brine chilled and the cooling water used 
were measured with meters, which were afterwards tested 
under the conditions of the experiment. 

It is interesting to compare the refrigerative effects ex- 
pressed in pounds of ice per pound of coal. On this basis the 
compression-machine tested by Professor Denton has an ad- 
vantage of 

24.1 — 17. 1 

X 100= iq per cent. 

24.1 ^ ^ v 

But this comparison is really unfair to the compression-ma- 
chine, for its steam-engine is assumed to require a consump- 
tion of three pounds of coal per horse-power per hour, while 
the calculation for the absorption-machine is based on the as- 
sumption that a pound of coal can evaporate ten pounds of 
water; but an automatic condensing-engine of the given 
power should be able to run on 20 or 22 pounds of steam per 
horse-power per hour. 



INDEX. 



PAGE 

Absolute temperature 32, 57, 71 

Absorption refrigerating apparatus 496 

Adiai atic for gases 66 

for liquid and vapor 117, 120 

for superheated vapor 135 

lines 19 

Air-compression, efficiency 455 

Air-compressor, calculation 459 

compound 451 

cooling during compression 444 

effect of clearance 447 

friction 454 

fluid-piston 443 

moisture in cylinder 445 

power expended 446 

three-stage 452 

Air, flow of 154 

friction in pipes 463 

pump 457 

thermometer 71 

Alternative method 49 

Ammonia 145 

Augsburg, engine test 356, 357 

Automatic and throttle engines 420 

Automatic-engine, tests .... 363, 381, 382 

Bache, tests on 388 

Barrel-calorimeter 289 

Bell-Coleman refrigerating machine 498 

Boston Main Drainage, engine test 360 

Boyle's law 54 

British thermal unit 7 

Brookline, tests on 358 

Callendar and Nicolson 343 

Calorie 15 

513 



5H INDEX. 

PAGE 

Calorimeter, barrel 289 

continuous 291 

separating 297 

throttling 294 

Carnot's engine 25 

function 31 

principle 29 

Characteristic equation , « 2 

for gases 55 

for superheated vapors - ■ 126, 130 

Chestnut Hill, engine test 355 

Coal-gas analysis 210 

Coefficient of dilatation 55 

Colchester, test on 358 

Compound air-compressor. « 451 

air-engine 470 

Compound-engine 255 

cross-compound 268 

direct-expansion 263 

gain from compounding 413 

indicator diagrams 262 

low-pressure cut-off 260 

ratio of cylinders • 61 

total expansions 259 

with receiver 258 

without receiver 257 

Compressed-air 442 

effect of clearance 447 

interchange of heat 449 

storage of power 475 

temperature after compression 448 

transmission of power 474 

Compressed-air engine 467 

calculation 471 

compound 470 

consumption 468 

final temperature 468 

interchange of heat 469 

moisture in cylinder 470 

power of 467 

volume of cylinder 469 

Condensation in high-speed engines 424 

Condensers 249 

cooling surface 251 

ejector 193 

Cornell, tests on engine 396 



INDEX. 5 I 5 

PAGE 

Crensot, tests on engine 381, 382 

Critical temperature no 

Curve of constant steam weight in 

Cut-off and expansion 417 

Cycle, closed 28 



non-reversible, 



42 

reversible 27 

Dallas, tests on 388 

Dean 359, 360 

Delafond 381, 382, 435 

Density of gases 59 

of mercury 58 

of vapors 107 

Denton 361, 400, 509, 510 

Designing steam-engines 252, 277 

Dexter, tests on 3S8 

Diesel motor 221 

Differential coefficient dp/dt g! 

Dilatation, coefficient of 55 

Direct-acting pumps, tests 364, 365 

Dixwell's tests 37! 

Dynamometers 284 

Effect of raising steam-pressure 248, 366 

Efficiency • 29 

mechanical 430 

of compressed-air transmission 474 

of reversible engines 35 

of steam-engine. . . . 230, 238, 240, 244 

Ejector 192 

condenser ig3 

Engine, Carnot's 25 

compressed-air 467 

Ericsson's igg 

friction of ' 429 

gas 194 ( 200 

hot-air ig4 

oil 216 

reversible 27 

steam 229 

Stirling's 196 

Entropy 22 

due to evaporization 116 

expression for 38 

of a liquid 115 

of a liquid and vapor 117 

of gases 70 



5 i6 



INDEX. 



Entropy of superheated vapor . 

scale of 

Ericsson's hot-air engine , 

Eutaw, tests on t 

Exhaust-steam injector 

Expansion and cut-off , 

Exponential equation 

First law of thermodynamics 

application of 

application to superheated vapors 

application to vapors 

Flow of air, Fliegner's equations 

in pipes 

maximum velocity 

through porous plug 

Flow of fluids 

of gases 

of incompressible fluids 

of saturated vapor 

of steam, experiments 

of superheated vapor 

Fluid-piston compressor 

French and English units 

Friction of engines 

distribution 

initial and load 

Fundamental equations 

Fusi Yama 

Gallatin, tests on 

Gas-engine 

governors 

ignition 

Otto 

tests 2IO, 212 

with compression in cylinder. 

with separate compression, 

Gas-producers 

Gases 

adiabatic equations , . . . 

density 

entropy 

flow of 

general equations , 

intrinsic energy = 

isoenergic equation 

isothermal equation 



PAGE 
128 

34 
199 

369 
]86 

417 
69 

15 

44 

125 

103 

154 
463 
155 
72 
149 

151 
150 
157 
159 
160 

443 

57 

429 

439 
431 
44 
358 
388 
200 
220 
218 
209 

215 
204 
201 
214 

54 
66 

59 
70 

151 
63 
68 

65 
64 



INDEX. 5 1 7 

PAGE 

Gases specific heats 62 

specific volumes 59 

Gauges 283 

Gay-Lussac's law 54 

General equations for gases 63 

for superheated vapors 127 

for vapors 102 

General principles, first 2 

second 15 

third 30 

Graphical representation of change of energy 20 

of characteristic equation 6 

of efficiency 36 

Gravity, acceleration of 57 

Hall's investigations 341 

Hallauer's tests . . = 319 

Heat of the liquid gy t g8 

Heat of vaporization 100 

Hirn engine, tests on 320 

Hirn's analysis 304 

experiments on superheated steam 134 

Hoadley engine, tests on 364 

Holyoke, tests on engine : 362 

Hot-air engines 194 

Indicators 284 

Influence of cylinder walls 301 

Callender and Nicolson 343 

Hall 341 

Hirn's analysis 304 

Initial condensation 301 

Injecter 163 

automatic iyg 

combining-tube . 176 

delivery-tube 176 

double 179 

efficiency of 177 

exhaust steam 1S6 

lifting 178 

Korting 180 

restarting. 183 

Schaffer and Budenberg 187 

Seller's 178 

Seller's experiments 182, 186 

steam-nozzle 175 

tests on 182, 186, 189 

theory 165 



5 J 8 INDEX. 



PAGE 



Injector velocity in delivery tube ^ni 

velocity of steam-jet j6g 

water I9 o 

Interchanges of heat, air-compressors 440 

compressed-air-engine 460 

steam-engine 301 

Intermediate rcheaters 4 OI 

Internal latent heat of vapors 102 

Intrinsic energy xy 

of gases 68 

of superheated vapors 132 

of vapors uq 

Iona, tests on..." 358 

Isherwood 363, 370 

Isoenergic or isodynamic line to 

for gases 65 

for superheated vapors 138 

for vapors 113 

isothermal lines ... t8 

for gases 64 

for superheated vapors 138 

for vapors , 112 

Joule and Thomson's experiments 2 

Joule's equivalent 15, 

Kennedy. 358 

Kilogram 57 

Kneass 168 

Laketown engine test 390 

Latent heat of expansion 9 

Latitude, standard 58 

Laws of thermodynamics : 15, 30 

application of 44, 47 

application to gases 61 

application to superheated vapors 124 

application to vapors 103 

Leavitt engines. 354, 355, 359, 360 

Linde refrigerating machine 502, 506 

Line of constant steam weight in 

Lines, adiabatic 19, 66, 117, 135 

isoenergic. < 19, 65, 113, 138 

isothermal . 18, 64, 112, 138 

of equal pressure 18, 112 

of equal volume . . . . , 18 

Louisville, test on engine 359 

Marine engine tests 358, 386 

Mass. Inst. Technology, engine tests < 371, 389, 401 



INDEX. 519 



PAGE 



Mechanical efficiency 430 

Mechanical equivalent of heat 15, 95 

Joule's 96 

Rowland's 95 

Mercury, density of 88 

Meteor, test on 358 

Meter 88 

Michigan, tests on 302 

Miller, E. F 355 

Minneapolis, tests of auxiliary machinery 365 

Natick, engine test 360 

New Bedford, engine test 361 

Non-reversible cycles 42 

Oil-engines 216 

Perot's experiments on density of vapors 109 

Pictet's fluid 495 

refrigerating machine 502, 507 

Porus plug, flow through 72 

Pressure of saturated steam , „ . . 86, 88 

of vapors 80, 90 

specific 3 

Rankine's equations for flow of steam 158 

for pressure of steam 89 

Ratio of cylinders, compound engines 261 

Refrigerating machines 479 

absorption 496 

air 480 

calculations for 488, 494 

compression 490 

extraction of moisture 481 

fluids, for 495 

proportions 483, 491 

tests 499 

vacuum 498 

Regnault's equations for steam 84 

Relations of thermal capacities 10 

Revenue steamers, tests on 386 

Reversible cycle 27 

engine 27 

engine, efficiency 35 

Rontgen's experiments 76 

Rowland's experiments 93 

equivalent 95 

reduction of air-thermometer 96 

Rush, tests on « 388 

Saturated vapors 80 



520 INDEX, 

PAGE 

Saturated vapors adiabatic equations 117 

density 107 

entropy. . . 116 

flow of 157 

general equation 102 

intrinsic energy 113 

isoenergic equation 113 

isothermal equation 112 

pressure of 86, 88, 91 

specific heats 105 

specific volumes 107 

Schmidt's engines 375 

Schroter's tests of refrigerating machines 500 

tests of steam engines 357 

Seaton's multipliers for steam-engine design 253, 278 

Second law of thermodynamics 25 

application of 47 

application to superheated vapors .... 125 

application to vapors 104 

Sound, velocity of , 73 

Specific-heat 8 

of gases 62 

of liquids 98 

of steam 105 

of superheated steam 129, 132 

of vapors 105 

of water 97 

Specific-pressure 3, 58 

Specific-volume 3 

of gases 59 

of liquids 100 

of vapors 107 

Steam, curve of constant weight in 

flow of 157 

pressure of 86, 88 

Steam-engine 229 

actual 241 

Carnot's cycle . 229 

compound 255 

designing 252, 277 

economy ->...> . 353 

efficiency ■ - 244 

Hirn's analysis 304, 313, 323, 332, 336 

indicators 284 

influence of the cylinder walls .'.,.. 301 

leakage of valves 350 



INDEX. 521 

PAGE 

Steam-engine Seaton's multipliers 253, 278 

steam jackets 322 

superheated steam 319 

testing of 280 

tests of 353 

triple-expansion 259 

with non-conducting cylinder 235 

Steam-jackets 377 

gain from 400 

Steam-turbine 425 

Stirling's hot-air engine 196 

Storage of power, compressed air 475 

Sulphur dioxide 139 

Superheated vapors « 123 

adiabatic equation 135 

application of laws of thermodynamics. 124 

characteristic equation 130 

entropy 128 

flow of 160 

isothermal line 138 

specific heat 129, 132 

total h eat 133 

values of constants . 130 

Temperature. , 4 

absolute scale 32 

standard 7 

Temperature-entropy diagram 37 

Testing steam-engines 280 

indirect method 298 

Test of blowing-engine 457 

Tests of refrigerating machines, air 499 

- absorption 510 

compression 500 

Tests of steam-engines 353 

compound engines 359, 389, 396 

examples of economy 354 

marine engines 358, 386 

simple engines 318, 363, 381, 396 

steam -pumps 364, 365 

superheated steam 368 

triple engines 355, 389, 396 

Willans's engine 402 

Thermal capacities. 7 

of gases 63 

of superheated vapors 126 

relations of . . 10 



522 INDEX. 

PAGE 

Thermal lines 18 

Thermal unit 7 

Thermometers 283 

Thomson and Joule's experiments 72 

Thomson's scale of temperature 32 

Throttling and automatic engines 420 

Throttling calorimeter ■. 294 

Thurston v .. . 209, 438 

Total heat of steam 99 

of superheated steam 133 

of vapors 100 

Triple expansion engine, diagrams 273 

Vacuum refrigerating apparatus 498 

Value of R 60 

Velocity of sound 73 

Ville de Douvres 358 

Water injector 190 

Weirs 288 

Willans's engine 402 

Zeuner's equations for internal heat no 

Zeuner's general equations 51 



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1 



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Searles's Elements of Geometry 8vo, 

Totten's Metrology 8vo, 

Warren's Descriptive Geometry 2 vols., 8vo, 

' ' Drafting Instruments 12mo, 

" Free-hand Drawing 12mo, 

" Higher Linear Perspective 8vo, 

" Linear Perspective 12mo, 

" Primary Geometry 12mo, 

11 



1 50 


1 50 


1 50 


4 00 


1 00 


1 50 


1 50 


5 00 


1 50 


1 75 


1 50 


1 00 


3 50 


1 50 


1 50 


2 00 


3 00 


1 50 


5 00 


2 00 


2 50 


2 50 


3 00 


1 50 


1 50 


2 50 


3 50 


1 25 


1 00 


3 50 


1 00 


75 



Warren's Plane Problems. , 12mo, $1 25 

' ' Problems and Theorems 8vo, 2 50 

" Projection Drawing 12mo, 1 50 

Wood's Co-ordinate Geometry 8vo, 2 00 

" Trigonometry 12mo, 1 00 

Woolf s Descriptive Geometry Royal 8vo, 3 00 

MECHANICS-MACHINERY. 

Text-books and Practical Works. 
(See also Engineering, p. 6.) 

Baldwin's Steam Heating for Buildings 12mo, 

Benjamin's Wrinkles and Recipes 12mo, 

Carpenter's Testing Machines and Methods of Testing 

Materials 8vo. 

Chordal's Letters to Mechanics . 12mo, 

Church's Mechanics of Engineering 8vo, 

" Notes and Examples in Mechanics 8vo, 

Crehoi e's Mechanics of the Girder 8vo, 

Cromwell's Belts and Pulleys. 12mo, 

'■ Toothed Gearing 12mo, 

Compton's First Lessons in Metal Working 12mo, 

Dana's Elementary Mechanics 12mo, 

Dingey's Machinery Pattern Making 12mo, 

Dredge's Trans. Exhibits Building, World Exposition, 

4to, half morocco, 

Du Bois's Mechanics. Vol. I., Kinematics 8vo, 

Vol. II.. Statics 8vo, 

Vol III., Kinetics 8vo, 

Fitzgerald's Boston Machinist 18mo, 

Flather's Dynamometers 12mo, 

" Rope Driving 12mo, 

Hall's Car Lubrication 1 2mo, 

Holly's Saw Filing 18mo, 

Johnson's Theoretical Mechanics. An Elementary Treatise. 
(In the press. ) 

Jones Machine Design. Part I., Kinematics 8vo, 1 50 

" " " Part II., Strength and Proportion of 

Machine Parts. 

Lanza's Applied Mechanics 8vo, 7 50 

MacCord's Kinematics 8vo, 5 00 

Merriman's Mechanics of Materials « 8vo, 4 00 

Metcalfe's Cost of Manufactures 8vo, 5 00 

Michie's Analytical Mechanics Svo, 4 00 

Mosely's Mechanical Engineering. (Mahan.) 8vo. 5 00 

12 



2 50 


2 00 


2 00 


6 00 


2 00 


5 00 


1 50 


1 50 


1 50 


1 50 


2 00 


10 00 


3 50 


4 00 


3 50 


1 00 


2 00 


2 00 


1 00 


75 



Richards's Compressed Air 12mo, $1 50 

Robinson's Principles of Mechanism 8vo, 3 00 

Smith's Press- working of Metals .8vo, 3 00 

The Lathe and Its Uses 8vo, 6 00 

Thurston's Friction and Lost Work 8vo, 3 00 

" The Animal as a Machine 12mo, 1 00 

"Warren's Machine Construction 2 vols., 8vo, 7 50 

Weisbach's Hydraulics and Hydraulic Motors. (Du Bois.)..8vo, 5 00 
" Mechanics of Engineering. Vol. III., Part I., 

Sec. I. (Klein.) 8vo, 5 00 

"Weisbach's Mechanics of Engineering Vol. III., Part I., 

Sec. II (Klein.) 8vo, 5 00 

"Weisbach's Steam Engines. (Du Bois.) 8vo, 5 00 

Wood's Analytical Mechanics 8vo, 3 00 

' ' Elementary Mechanics ........ 12rao, 1 25 

" " " Supplement and Key 1 25 



METALLURGY. 

Iron — Gold— Silver — Alloys, Etc. 

Allen's Tables for Iron Analysis 8vo, 

Egleston's Gold and Mercury 8vo, 

" Metallurgy of Silver , 8vo, 

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* " " Steel, Fuel, etc 8vo, 

Kunhardt's Ore Dressing in Europe 8vo, 

Metcalf's Steel — A Manual for Steel Users 12mo, 

O'Driscoll's Treatment of Gold Ores 8vo, 

Thurston's Iron and Steel 8vo, 

Alloys 8vo, 

Wilson's Cyanide Processes 12mo, 

MINERALOGY AND MINING. 

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Barringer's Minerals of Commercial Value. .. .oblong morocco, 2 50 

Beard's Ventilation of Mines 12mo, 2 50 

Boyd's Resources of South Western Virginia 8vo, 3 00 

" Map of South Western Virginia Pocket-book form, 2 00 

Brush and Penfield's Determinative Mineralogy 8vo, 3 50 

Chester's Catalogue of Minerals 8vo, 1 25 

paper, 50 

" Dictionary of the Names of Minerals 8vo, 3 00 

Dana's American Localities of Minerals 8vo, 1 00 

13 



3 


00 


r 
1 


50 


7 50 


5 


00 


5 


00 


1 


50 


2 


00 


2 00 


3 


50 


2 


50 


1 


50 



Dana's Descriptive Mineralogy. (E. S.) . . . .8vo, half morocco, $12 50 

" Mineralogy and Petrography (J.D.) 12mo, 2 00 

" Minerals and How to Study Them. (E. S.) 12mo, 1 50 

" Text-book of Mineralogy. (E. S.) 8vo, 3 50 

*Drinker's Tunnelling, Explosives, Compounds, and Rock Drills. 

4to, half morocco, 25 00 

Egleston's Catalogue of Minerals and Synonj^ms 8vo, 2 50 

Eissler's Explosives — Nitroglycerine and Dynamite 8vo, 4 00 

Goodyear 's Coal Mines of the Western Coast 12mo, 2 50 

Hussak's Rock forming Minerals (Smith.) 8vo, 2 00 

Ihlseng's Manual of Mining . . . . 8vo, 4 00 

Kunhardt's Ore Dressing in Europe < 8vo. 1 50 

O'Driscoll's Treatment of Gold Ores . . . 8vo, 2 00 

Rosenbusch's Microscopical Physiography of Minerals and 

Rocks (Iddings ) 8vo, 5 00 

Sawyer's Accidents in Mines. . . 8vo, 7 00 

Stockbridge's Rocks and Soils. . , , ,. . . .8vo, 2 50 

Walke's Lectures on Explosives 8vo, 4 00 

Williams's Lithology 8vo, 3 00 

Wilson's Mine Ventilation 161110, 125 

" Hydraulic and Placer Mining 12mo. 

STEAM AND ELECTRICAL ENGINES, BOILERS, Etc. 

Stationary — Marine— Locomotive — Gas Engines, Etc. 

(See also Engineering, p. 6.) 

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Clerk's Gas Engine t 12mo. 

Ford's Boiler Making for Boiler Makers 18mo, 

Hemenway's Indicator Practice 12mo. 

Hoadley's Warm-blast Furnace 8vo, 

Kneass's Practice and Theory of the Injector 8vo, 

MacCord's Slide Valve 8vo, 

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Meyer's Modern Locomotive Construction 4to, 

Peabody and Miller's Steam Boilers 8vo, 

Peabody's Tables of Saturated Steam 8vo, 

" Thermodynamics of the Steam Engine 8vo, 

" Valve Gears for the Steam Engine 8vo, 

Pray's Twenty Years with the Indicator Royal 8vo, 

Pupin and Osterberg's Thermodynamics 12mo, 

Reagan's Steam and Electrical Locomotives 12mo, 

Rontgen's Thermodynamics. (Du Bois.) 8vo, 

Sinclair's Locomotive Running 12mo, 

Thurston's Boiler Explosion. 12mo,' 

14 



2 50 


4 00 


1 00 


2 00 


1 50 


1 50 


2 00 


18 00 


10 00 


4 00 


1 00 


5 00 


2 50 


2 50 


1 25 


2 00 


5 00 


2 00 


1 50 



Thurston's Engine and Boiler Trials 8vo, $5 00 

" Manual of the Steam Engine. Part L, Structure 

and Theory. 8vo, 7 50 

Manual of the Steam Eugine. Part II. , Design, 

Construction, and Operation 8vo, 7 50 

2 parts, 12 00 

" Philosophy of the Steam Engine 12mo, 75 

" Reflection on the Motive Power of Heat. (Caruot.) 

12mo, 1 50 

" Stationary Steam Engines 12mo, 1 50 

" Steam-boiler Coustruction and Operation 8vo, 5 00 

Spangler's Valve Gears 8vo, 2 50 

Trowbridge's Stationary Steam Engines 4to, boards, 2 50 

Weisbach's Steam Engine. (DuBois.) 8vo, 5 00 

Whitham's Constructive Steam Engineering 8vo, 10 00 

' ' Steam-engine Design 8vo, 5 00 

Wilson's Steam Boilers. (Flather.) 12mo, 2 50 

Wood's Thermodynamics, Heat Motors, etc 8vo, 4 00 

TABLES, WEIGHTS, AND MEASURES. 

For Actuaries, Chemists, Engineers, Mechanics— Metric 

Tables, Etc 

Adriance's Laboratory Calculations 12mo, 1 25 

Allen's Tables for Iron Analysis 8vo, 3 00 

Bixby's Graphical Computing Tables Sheet, 25 

Compton's Logarithms 12mo, 1 50 

Crandall's Railway and Earthwork Tables. 8vo, 1 50 

Egleston's Weights and Measures 18mo, 75 

Fisher's Table of Cubic Yards Cardboard, 25 

Hudson's Excavation Tables. Vol. II 8vo, 1 00 

Johnson's Stadia and Earthwork Tables 8vo, 1 25 

Ludlow's Logarithmic and Other Tables. (Bass.) 12mo, 2 00 

Thurston's Conversion Tables 8vo, 1 00 

Totten's Metrology 8vo, 2 50 

VENTILATION. 

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Baldwin's Steam Heating 12mo, 2 50 

Beard's Ventilation of Mines 12mo, 2 50 

Carpenter's Heating and Ventilating of Buildings ,8vo, 3 00 

Gerhard's Sanitary House Inspection Square 16mo, 1 00 

Mott's The Air We Breathe, and Ventilation , 16mo, 1 00 

Reid's Ventilation of American Dwellings , . . . 12mo, 1 50 

Wilson's Mine Ventilation 16mo, 1 25 

15 



MISCELLANEOUS PUBLICATIONS. 

Alcott's Gems, Sentiment, Language . Gilt edges, $5 00 

Bailey's The New Tale of a Tub 8vo, 75 

Ballard's Solution of the Pyramid Problem 8vo, 1 50 

Barnard's The Metrological System of the Great Pyramid. ,8vo. 1 50 

Davis's Elements of Law 8vo, 2 00 

Emmon's Geological Guide-book of the Rocky Mountains. .8vo, 1 50 

Ferrel's Treatise on the Winds 8vo, 4 00 

Haines's Addresses Delivered before r the Am. Ry. Assn. . .12mo. 2 50 

Mott's The Fallacy of the Present Theory of Sound. .Sq. 16mo, 1 00 

Perkins's Cornell University Oblong 4to, 1 50 

Rickettsia History of Rensselaer Polytechnic Institute 8vo, 3 00 

Rctherham's The New Testament Critically Emphasized. 

12mo, 1 50 
" The Emphasized New Test. A new translation. 

Large 8vo, 2 00 

Totteu's An Important Question in Metrology 8vo, 2 50 

Whitehouse's Lake Mceris e ..... . Paper, 25 

* Wiley's Yosemite, Alaska, and Yellowstone 4to, 3 00 

HEBREW AND CHALDEE TEXT=BOOKS. 

For Schools and Theological Seminaries. 

Gesenius's Hebrew and Chaldee Lexicon to Old Testament. 

(Tregelles. ) . Small 4to, half morocco, 5 00 

Green's Elementary Hebrew Grammar 12mo, 1 25 

" Grammar of the Hebrew Language (New Edition ).8vo, 3 00 

" Hebrew Chrestomathy 8vo, 2 00 

Letteris's Hebrew Bible (Massoretic Notes in Euglish). 

8vo ; arabesque, 2 25 
Luzzato's Grammar of the Biblical Chaldaic Language and the 

Talmud Babli Idioms 12mo, 1 50 

MEDICAL. 

Bull's Maternal Management in Health and Disease 12mo, 1 00 

Hammarsteu's Physiological Chemistry. (Mandel.) 8vo, 4 00 

Mott's Composition, Digestibility, and Nutritive Value of Food. 

Large mounted chart, 1 25 

Ruddiman's Incompatibilities in Prescriptions 8vo, 2 00 

Steel's Treatise on the Diseases of the Ox 8vo, 6 00 

Treatise on the Diseases of the Dog 8vo, 3 50 

Woodhull's Military Hygiene 12mo, 1 50 

Worcester's Small Hospitals— Establishment and Maintenance, 
including Atkinson's Suggestions for Hospital Archi- 
tecture 12mo, 1 25 

16 









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